If you write your system in the form AX = B, and A is invertible, then you can multiply both sides by the inverse of A to solve for X. Here A, X, and B are matrices of the variables' coefficients, the variables themselves, and the equations' constants. Show
First, you must be able to write your system in Standard form, before you write your matrix equation. As you know from other operations, the Identity produces itself (adding 0, multiplying by 1), leaving you with the variables alone on the left side, and your answers on the right! All you have to do is multiply matrix #A^-1# times matrix B. Here you can solve systems of simultaneous linear equations using Inverse Matrix Method Calculator with complex numbers online for free. All the auxiliary methods used in calculation can be calculated apart with more details. Have questions? Read the instructions. To solve
a system of linear equations using inverse matrix method you need to do the following steps. To understand inverse matrix method better input any example and examine the solution. In this explainer, we will learn how to solve a system of three linear equations using the inverse of the matrix of coefficients. We can solve a system of linear equations, which are also called simultaneous equations, using the substitution or elimination methods, but these methods become convoluted when the number of equations are more than two. Even with a system of three equations, this process is time consuming to solve by hand. But if we want to program a computer to
perform this task for us, we need a more systemic approach to this task. This is where the matrix method comes in. One of the most widely used applications of matrix operations is to formalize this process by means of the matrix inverse so that we can easily program a computer to perform this task. We will see further below in this explainer how to write these systems of
๐ linear equations as one matrix equation of the form ๐ด๐
=๐ต, where ๐ด is a square matrix of order ๐ร๐ and ๐ and ๐ต are matrices of order ๐ร1.
๐ is the unknown matrix (i.e., its ๐ elements are unknown). Let us begin first by discussing how to solve a matrix equation of the form
๐ด๐=๐ต using the matrix inverse. We know that ๐ด is a square matrix. Recall that the inverse of a square matrix exists if its determinant is not equal to zero. Given a 3ร3 matrix
๐ด with det๐ดโ 0, the inverse matrix ๐ด๏ฑ๏ง
is the 3ร3 matrix satisfying ๐ด๐ด=๐ด๐ด=๐ผ,๏ฑ๏ง๏ฑ๏ง
where ๐ผ is the 3ร3
identity matrix. Now, to solve the matrix equation ๐ด๐=๐ต, where
๐ด and ๐ต are known
3ร3 and 3ร1 matrices, respectively, we need to multiply from the left by ๐ด๏ฑ๏ง on both sides of the equation to obtain
๐ด๐ด๐=๐ด๐ต.๏ฑ๏ง๏ฑ๏ง Since ๐ด๐ด=๐ผ๏ฑ๏ง, this equation simplifies to ๐=๐ด๐ต.
๏ฑ๏ง Both ๐ด๏ฑ๏ง and
๐ต are known matrices; hence, this gives the solution to the matrix equation ๐ด๐=๐ต. Let ๐ด be an invertible matrix and ๐ต be a matrix such that the multiplication ๐ด๐ต๏ฑ๏ง is defined. Matrix ๐ satisfying the equation ๐ด๐=๐ต is given by ๐=๐ด๐ต.๏ฑ๏ง This method gives us a way to solve any matrix equation of the form ๐ด๐=๐ต
if matrix ๐ด is invertible. However, this method cannot be used when ๐ด is not
invertible. This could happen if ๐ด is not a square matrix or if ๐ด is square and
det๐ด=0. In such cases, the matrix equation has either an infinite
number of solutions or no solution. We will not focus on these scenarios in this explainer, and we will check that the coefficient matrix is invertible before proceeding. In our first example, we will solve a matrix equation when the inverse of a 3ร3 matrix is provided. Given that ๏113025301๏=๏โ211โ15856โ3โ2๏,๏ฑ
๏ง solve the following matrix equation for ๐: ๏
12370โ102โ2๏โ๏113025301๏๐=๏โ1226โ112โ20๏. In this example, we need to solve a matrix equation to find the unknown matrix ๐. To solve this equation, we want to rearrange the equation so that
๐ is the subject. We can begin by subtracting both sides of the equation by the leftmost matrix in the equation: โ๏113025301๏๐=๏โ1226โ112โ20๏โ๏12370โ102โ2๏=๏โ20โ1โ1โ122โ42๏. Now, we can multiply both sides of the equation by โ1 to write ๏11302
5301๏๐=๏20111โ2โ24โ2๏. Finally, we can multiply from the left by the provided inverse matrix on both sides of the equation to write ๏โ211โ1585 6โ3โ2๏๏113025301๏๐=๏โ211โ15856โ3โ2๏๏20111โ2โ24โ2๏. We know that, for any invertible square matrix ๐ด, ๐ด๐ด=๐ผ,๏ฑ๏ง where ๐ผ is the identity matrix of the same order. Hence, the multiplication of the two matrices on the left-hand side of the equation will result in the identity matrix, which simplifies the equation to ๐=๏โ21 1โ15856โ3โ2๏๏20111โ2โ24โ2๏. Hence, we can finish by working out the matrix multiplication: ๐=๏โ2ร2+1ร1+1ร(โ2)โ2ร0+1ร1+1ร4โ2ร1+1ร(โ2)+1ร(โ2)โ15ร2+8ร1+5ร(โ2)โ15ร0+8ร1+5ร4โ15ร1+8ร(โ2)+5ร(โ2)6ร2+(โ3)ร1+(โ2)ร(โ2)6ร0+(โ3)ร1+(โ2)ร46ร1+(โ3)ร(โ2)+(โ2)ร(โ2)๏=๏โ55โ6โ3228โ4113โ1116๏. In the previous example, we solved a matrix equation using a matrix inverse. However, we were given the inverse of the 3ร3 matrix, which is usually the most difficult part. If we are not given the inverse matrix, we will need to first find the inverse matrix. Let us recall the adjoint method for finding the inverse of a 3ร3 matrix. How To: Finding the Inverse of a 3 ร 3 Matrix with the Adjoint MethodFor a 3ร3 matrix ๐ด with det๐ดโ 0, we can find the inverse matrix ๐ด๏ฑ๏ง by the following steps:
As we can see above, finding the inverse of a 3ร3 matrix is a tedious process. The same method can be used for square matrices of higher order; it would be too lengthy to even compute the determinant of the matrix by hand, let alone the inverse. For this reason, many scientific calculators or mathematical programs have built-in functions for computing the inverse of a matrix. For a 3ร3 matrix, we can compute the inverse matrix by hand using the adjoint method. In the next example, we will find the inverse of a 3ร3 matrix using the adjoint method and use it to solve a given matrix equation. Example 2: Solving a Matrix Equation by Finding the Inverse of a MatrixSolve ๏1โ1โ111โ 1110๏๏ฟ๐ฅ๐ฆ๐ง๏=๏9โ 116๏ using the inverse of a matrix. AnswerIn this example, we need to solve a matrix equation. To solve this equation, we need to multiply from the left by the inverse of the given 3ร3 matrix on both sides of the equation. Let us begin by finding the inverse of the 3ร3 matrix: ๐ด=๏1โ1โ11 1โ1110๏. Recall that a square matrix is invertible if its determinant is nonzero. Let us begin by computing the determinant of this matrix and making sure that it is nonzero. We recall that, for a 3ร3 matrix ๐ด=๏น๐๏ ๏๏ , its determinant can be computed by det๐ด=๐|๐ด|โ๐| ๐ด|+๐|๐ด|,๏ง๏ง๏ง๏ง ๏ง๏จ๏ง๏จ๏ง๏ฉ๏ง๏ฉ where ๐ด๏๏ are matrix minors obtained by taking the ๐th row and ๐th column from matrix ๐ด. We can apply this formula to our coefficient matrix ๐ด to obtain det๐ด=1ร| |1โ110||โ(โ1)ร||1โ110||+(โ1)||1111||=1(1ร0โ(โ1)ร1)โ(โ 1)(1ร0โ(โ1)ร1)+(โ1)(1ร1โ1ร1)=1+1+0=2. Since det๐ดโ 0, we know that the inverse matrix ๐ด๏ฑ๏ง exists. We can find the inverse matrix by using the adjoint method as follows:
Let us first find the cofactor matrix. Entries of the cofactor matrix are the determinants of the corresponding matrix minors multiplied by the alternating sign (โ1)๏๏ฐ๏ . We need to compute the determinants of 9 matrix minors with the corresponding sign for this purpose: +|๐ด|=+||1โ110||=1,โ|๐ด|=โ||1โ110||=โ1,+|๐ด|=+||1111||=0,โ|๐ด|=โ||โ1โ110||=โ1,+|๐ด|=+||1โ110||=1,โ|๐ด|=โ||1โ111||=โ2,+|๐ด|=+||โ1โ11โ1||=2,โ| ๐ด|=โ||1โ11โ1| |=0,+|๐ด|=+||1โ111||=2.๏ง๏ง๏ง ๏จ๏ง๏ฉ๏จ๏ง๏จ๏จ๏จ๏ฉ๏ฉ๏ง๏ฉ๏จ๏ฉ๏ฉ This leads to the cofactor matrix ๏1โ10โ 11โ2202๏. We can find the adjoint matrix by taking the transpose: adj๐ด=๏1โ 12โ1100โ22๏. Finally, by multiplying by the reciprocal of the determinant, which we computed earlier to be 2, we obtain ๐ด=12๏1โ12โ1100โ22๏.๏ฑ๏ง Now that we have found the inverse matrix, we can multiply this matrix from the left on both sides of the given equation to write 12๏1โ12โ1100โ22๏๏1โ1โ111โ1110๏๏ฟ๐ฅ๐ฆ๐ง๏=12๏1โ12โ1100โ22๏๏9โ116๏. We know that, for any invertible square matrix ๐ด, ๐ด๐ด=๐ผ,๏ฑ๏ง where ๐ผ is the identity matrix of the same order. Hence, the multiplication of the two matrices and the scalar on the left-hand side of the equation will result in the identity matrix, which simplifies the equation to ๏ฟ๐ฅ๐ฆ๐ง๏=12๏1โ12โ1100 โ22๏๏9โ116๏. Hence, we can finish by computing the matrix and scalar multiplication on the right-hand side of this equation: ๏ฟ๐ฅ๐ฆ๐ง๏=12๏1ร9+(โ1)ร(โ11)+2ร6โ1ร9+1ร(โ11)+0ร60ร9+(โ2)ร(โ11)+2ร6๏=12๏32โ2034๏=๏16โ1017๏. This leads to ๏ฟ๐ฅ๐ฆ๐ง๏=๏16โ1017๏. Equating the corresponding entries of the matrices above, we obtain ๐ฅ=16,๐ฆ=โ10,๐ง=17. In the previous example, we solved a given matrix equation by first finding the inverse of a 3ร3 matrix. The matrix equation that we solved in this example is equivalent to a system of 3 equations with 3 unknowns. Once we write a system of equations into its matrix form, we can follow this method to solve the system of equations. Let us recall how to write a matrix equation equivalent to a given system of linear equations. Definition: Matrix Form of a System of Linear EquationsConsider a general system of linear equations with unknown variables ๐ฅ,๐ฅ,โฆ,๐ฅ๏ง ๏จ๏: ๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐,๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐,โฎโฎโฎโฑโฎโฎโฎ๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐.๏ง๏ง๏ง๏ง ๏จ๏จ๏ง๏๏๏ง๏จ๏ง๏ง๏จ๏จ๏จ๏จ๏๏๏จ๏๏ง๏ง๏๏จ๏จ๏๏๏๏ The coefficient matrix ๐ด is defined by ๐ด=โโโโ๐๐โฏ๐๐๐โฏ๐โฎโฎโฑโฎ๐๐โฏ๐โโโโ . ๏ง๏ง๏ง๏จ๏ง๏๏จ๏ง๏จ๏จ๏จ๏๏๏ง๏๏จ๏๏ Also, the variable and constant matrices ๐ and ๐ต, respectively, are given by ๐=โโโโ๐ฅ๐ฅโฎ๐ฅโโโโ ,๐ต=โโโโโ๐๐โฎ๐โโโโโ .๏ง๏จ๏๏ง๏จ๏ The given system of linear equations is equivalent to the matrix equation ๐ด๐=๐ต. We can see that the number of rows in the coefficient matrix ๐ด is equal to the number of equations, and the number of its columns is equal to the number of unknown variables. Hence, if we begin with a system of three equations containing three unknowns, the order of the coefficient matrix ๐ด will be 3ร3. This means that we need to find the inverse of a 3ร3 matrix in order to solve this matrix equation. In the next example, we will write a matrix equation that is equivalent to a given system of 3 linear equations and 3 unknowns. We will then solve the matrix equation using the matrix inverse. Example 3: Solving a Set of Simultaneous Equations Using MatricesConsider the system of equations 2๐+2 ๐+4๐=4,โ๐โ๐โ๐ =14,2๐+5๐+6๐=10.
AnswerPart 1 In this part, we need to write a matrix equation that is equivalent to the given system of 3 equations. Recall that a system of linear equations ๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐,๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐,โฎโฎโฎโฑโฎโฎโฎ๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐.๏ง๏ง๏ง๏ง๏จ๏จ ๏ง๏๏๏ง๏จ๏ง๏ง๏จ๏จ๏จ๏จ๏๏๏จ๏๏ง๏ง๏๏จ๏จ๏๏๏๏ is equivalent to the matrix equation โโโโ๐๐โฏ๐๐ ๐โฏ๐โฎโฎโฑโฎ๐๐โฏ๐โโโโ โโโโ๐ฅ๐ฅโฎ๐ฅโโโโ =โโโโโ๐๐โฎ๐โโโโโ .๏ง๏ง๏ง๏จ๏ง๏๏จ๏ง ๏จ๏จ๏จ๏๏๏ง๏๏จ๏๏๏ง๏จ๏๏ง๏จ๏ The matrices in the equation above are called the coefficient, variable, and constant matrices, respectively. From the given system of equations, our variables have names ๐ , ๐, and ๐, which form the entries of the variable matrix. The constants 4, 14, and 10 on the right-hand sides of the given equations form the entries of the constant matrix. Hence, the variable and constant matrices are, respectively, ๏๐๐๐ ๏,๏41410๏. To find the coefficient matrix, we need to write down the coefficients of each variable in the correct order (that is, the order of ๐, ๐, and ๐) for each equation. The coefficients are not explicitly visible in the second equation, since only the negative signs appear in front of the variables. This indicates that the coefficients of ๐, ๐, and ๐ in the second equation are โ1. We can write this into the equations: 2๐+2๐+4๐= 4,โ1๐โ1๐โ1๐=1 4,2๐+5๐+6๐=10 . This leads to the coefficient matrix ๏224โ1โ1โ1256๏. Hence, the matrix equation is ๏224โ1โ1โ125 6๏๏๐๐๐๏=๏41410๏. Part 2 In this part, we need to find the inverse of the coefficient matrix. We obtained, in the previous part, that the coefficient matrix is ๐ด=๏224โ1โ1โ1256๏. Recall that a square matrix is invertible if its determinant is nonzero. Let us begin by computing the determinant of this matrix and making sure that it is nonzero. We recall that, for a 3 ร3 matrix ๐ด=๏น๐๏ ๏๏ , its determinant can be computed by det๐ด=๐|๐ด|โ๐|๐ด|+๐|๐ด|๏ง๏ง๏ง๏ง๏ง๏จ๏ง๏จ๏ง๏ฉ๏ง๏ฉ where ๐ด๏๏ are matrix minors obtained by taking the ๐th row and ๐th column from matrix ๐ด. We can apply this formula to our coefficient matrix ๐ด to obtain det๐ด=2||โ1โ156||โ2||โ1โ126||+4||โ1โ125||=2((โ1)ร6โ(โ1)ร 5)โ2((โ1)ร6โ(โ1)ร2)+4((โ1)ร 5โ(โ1)ร2)=2ร(โ1)โ2ร(โ4)+4ร(โ3)=โ6. Since det๐ดโ 0, we know that the inverse matrix ๐ด๏ฑ๏ง exists. Let us find the inverse. Recall that we can find the inverse matrix by using the adjoint method as follows:
Let us first find the cofactor matrix. Entries of the cofactor matrix are the determinants of the corresponding matrix minors multiplied by the alternating sign (โ1)๏๏ฐ๏ . We need to compute the determinants of 9 matrix minors with the corresponding sign for this purpose: +|๐ด|=+||โ1โ156||=โ1,โ|๐ด|=โ||โ1โ126||=4,+|๐ด|=+||โ1โ125||=โ3,โ|๐ด|=โ||2456||=8,+|๐ด|=+||2426||=4,โ|๐ด|=โ||2225||=โ6,+|๐ด|=+||24โ1โ1||=2,โ|๐ด|=โ||24โ1โ1||=โ2, +|๐ด|=+||22โ1โ1||=0.๏ง๏ง๏ง๏จ๏ง๏ฉ ๏จ๏ง๏จ๏จ๏จ๏ฉ๏ฉ๏ง๏ฉ๏จ๏ฉ๏ฉ This leads to the cofactor matrix ๏โ14โ384โ 62โ20๏. We can find the adjoint matrix by taking the transpose: adj๐ด=๏โ18244โ2โ3โ60๏. Finally, by multiplying by the reciprocal of the determinant, which we computed earlier to be โ6, we obtain ๐ด=โ16๏โ18244โ2โ3โ60๏.๏ฑ๏ง Part 3 In this part, we need to solve the matrix equation by multiplying through by the inverse on the left-hand side. We recall the matrix equation we obtained in part 1: ๏224โ1โ1โ1256๏๏๐๐๐๏=๏41410๏. If we multiply from the left by the inverse matrix on both sides of the equation, we obtain โ16๏โ18244โ2โ3โ60๏๏224โ1โ1โ1256๏๏๐๐๐๏=โ16๏โ18244โ2โ 3โ60๏๏41410๏. We know that the two matrices on the left-hand side of the equation are inverses of each other, which means that their product will be the identity matrix. This simplifies this equation to ๏๐๐๐๏=โ16๏โ18244โ2โ3โ60๏๏41410๏. We can finish by computing the matrix multiplication on the right-hand side of the equation. This gives us ๏๐๐๐๏=โ16๏โ1ร4+8ร14+2ร104ร4+4ร14+(โ2)ร10โ3ร4+(โ6)ร14+0ร10๏=โ16๏12852โ96๏. Computing the scalar multiplication and simplifying, we obtain โ16๏12852โ 96๏=โโโโโโ643โ26316โโโโโ =13๏โ64โ2648๏. Hence, ๏๐๐๐๏ =13๏โ64โ2648๏. In the previous example, we wrote a matrix equation that is equivalent to a given system of three linear equations and solved the matrix equation using the matrix inverse. If we equate the corresponding entries of the solution of the matrix equation, we can find the solution of the system of equations. In the next example, we will solve a given matrix equation and find the unknown constants of the variable matrix. Example 4: Solving a System of Three Equation Using the Inverse of a MatrixUse the inverse of a matrix to solve the system of linear equations โ4๐ฅโ2๐ฆโ9๐ง=โ8,โ3๐ฅโ2๐ฆโ6๐ง=โ3,โ๐ฅ+๐ฆโ6๐ง=7. AnswerIn this example, we need to solve a system of 3 equations with 3 unknowns using matrices. We can begin by writing a matrix equation that is equivalent to the given system of equations. Recall that a system of linear equations ๐๐ฅ+๐๐ฅโฏ๐๐ฅ= ๐,๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐, โฎโฎโฎโฑโฎโฎโฎ๐๐ฅ+๐๐ฅโฏ ๐๐ฅ=๐.๏ง๏ง๏ง๏ง๏จ๏จ๏ง ๏๏๏ง๏จ๏ง๏ง๏จ๏จ๏จ๏จ๏๏๏จ ๏๏ง๏ง๏๏จ๏จ๏๏๏๏ is equivalent to the matrix equation โโโโ๐๐โฏ๐๐ ๐โฏ๐โฎโฎโฑโฎ๐๐โฏ๐โโโโ โโโโ๐ฅ๐ฅโฎ๐ฅโโโโ =โโโโโ๐๐โฎ๐โโโโโ .๏ง๏ง๏ง๏จ๏ง๏๏จ๏ง ๏จ๏จ๏จ๏๏๏ง๏๏จ๏๏๏ง๏จ๏๏ง๏จ๏ The matrices in the equation above are called the coefficient, variable, and constant matrices, respectively. From the given system of equations, our variables have names ๐ฅ , ๐ฆ, and ๐ง, which form the entries of the variable matrix. The constants โ8, โ3, and 7 on the right-hand sides of the given equations form the entries of the constant matrix. Hence, the variable and constant matrices are, respectively, ๏ฟ๐ฅ๐ฆ๐ง๏,๏โ8โ37๏. To find the coefficient matrix, we need to write down the coefficients of each variable in the correct order (that is, the order of ๐ฅ, ๐ฆ, and ๐ง) for each equation. In the final equation, the coefficients of ๐ฅ and ๐ฆ are not visible, which means that they are โ1 and 1, respectively. We can add these to the equation to write โ4๐ฅโ2๐ฆโ9๐ง=โ8,โ3๐ฅโ2๐ฆโ6๐ง=โ3,โ1๐ฅ+1๐ฆโ6๐ง=7. This leads to the coefficient matrix ๏โ4โ2โ9โ3โ2โ6โ11โ6๏. Hence, the matrix equation is
We can solve this equation by multiplying from the left the inverse of the coefficient matrix on both sides of the equation (1). Let us find the inverse of the coefficient matrix ๐ด=๏โ 4โ2โ9โ3โ2โ6โ11โ6๏. We can use the adjoint method to obtain the inverse of this matrix, if it exists. Recall that a square matrix is invertible if its determinant is nonzero. Let us begin by computing the determinant of this matrix and making sure that it is nonzero. We recall that, for a 3ร3 matrix ๐ด=๏น๐๏ ๏๏ , its determinant can be computed by d et๐ด=๐|๐ด|โ๐|๐ด |+๐|๐ด|๏ง๏ง๏ง๏ง๏ง๏จ ๏ง๏จ๏ง๏ฉ๏ง๏ฉ where ๐ด ๏๏ are matrix minors obtained by taking the ๐th row and ๐th column from matrix ๐ด. We can apply this formula to our coefficient matrix ๐ด to obtain det๐ด=(โ4)||โ2โ61โ6||โ(โ2) ||โ3โ6โ1โ6||+ (โ9)||โ3โ2โ11||=(โ4)ร18โ(โ2)ร12+(โ9)ร(โ5)=โ3. Since det๐ดโ 0, we know that the inverse matrix ๐ด๏ฑ ๏ง exists. Let us find the inverse. Recall that we can find the inverse matrix by using the adjoint method as follows:
Let us first find the cofactor matrix. Entries of the cofactor matrix are the determinants of the corresponding matrix minors multiplied by the alternating sign (โ1)๏๏ฐ๏ . We need to compute the determinants of 9 matrix minors with the corresponding sign for this purpose: +|๐ด|=+||โ2โ61โ6||=18,โ|๐ด|=โ||โ3โ6โ1โ6||=โ12,+|๐ด|=+||โ3โ2โ11||=โ5,โ|๐ด|=โ||โ2โ91โ6||=โ21,+|๐ด|=+||โ4โ9โ1โ6||=15,โ|๐ด|=โ||โ4โ2โ11||=6,+|๐ด|=+||โ2โ9โ2โ6||=โ6,โ|๐ด|=โ||โ4โ9โ3โ6||=3,+|๐ด|=+||โ4โ2โ3โ2||=2.๏ง๏ง๏ง๏จ๏ง๏ฉ๏จ๏ง๏จ๏จ๏จ๏ฉ๏ฉ๏ง๏ฉ๏จ๏ฉ๏ฉ This leads to the cofactor matrix ๏18โ12โ5โ 21156โ632๏. We can find the adjoint matrix by taking the transpose: adj๐ด=๏18 โ21โ6โ12153โ562๏. Finally, by multiplying by the reciprocal of the determinant, which we computed earlier to be โ3, we obtain ๐ด=โ13๏18โ21โ6โ12153โ562๏.๏ฑ๏ง Now that we have found the inverse matrix, let us multiply equation (1) through by the inverse on the left-hand side: โ13๏18โ21โ6โ12153โ562๏๏โ18โ34โ386โ1219๏๏ฟ๐ฅ๐ฆ๐ง๏=โ13๏18โ21โ6โ12153โ5 62๏๏โ8โ37๏. Since any matrix multiplied by its inverse results in the identity matrix, the two matrices and the scalar on the left-hand side of this equation cancel out. This simplifies the equation to ๏ฟ๐ฅ๐ฆ๐ง๏=โ13๏18โ21โ6โ12153โ562๏๏โ8โ37๏. Hence, we can finish by computing the matrix multiplication on the right-hand side of the equation above: ๏ฟ๐ฅ๐ฆ๐ง๏=โ13๏18ร(โ8)โ21ร(โ3)โ6ร7โ12ร(โ8)+15ร(โ3)+3ร7โ5ร(โ8)+6ร(โ3)+2ร7๏=โ13๏โ1237236๏=๏41โ24โ12๏. Hence, ๏ฟ๐ฅ๐ฆ๐ง๏ =๏41โ24โ12๏. Equating the corresponding entries in the matrices above, we obtain ๐ฅ=41,๐ฆ=โ 24,๐ง=โ12. In our final example, we will solve a real-world problem using the inverse of a 3ร3 matrix. Example 5: Solving a Real-World Problem Using Matrix InverseThe table below shows the number of different types of rooms in three hotels owned by a company.
All three hotels charge an equal amount for a room of the same size. When all the rooms are booked, the companyโs daily income from the first, second, and third hotels are 50โ โ120 LE, 53โ โ560 LE, and 55โ โ660 LE respectively. Find the daily income from a suite. AnswerIn this example, we have three unknown quantities: the costs of a single room, a double room, and a suite. Let us denote these unknowns by constants ๐ฅ, ๐ฆ, and ๐ง respectively. We can find the cost in LE of a suite by finding the value of ๐ง. We are given that the daily income from the first hotel is 50โ โ120 LE if all rooms are booked. This can be written as the following equation: 45๐ฅ+74๐ฆ+15๐ง=50120. Similarly, we can obtain another two equations from the daily income of the second and third hotels respectively: 48๐ฅ+7 4๐ฆ+19๐ง=53560, 49๐ฅ+94๐ฆ+10๐ง=5 5660. This gives us a system of three equations with three unknowns. Let us solve this system using matrices. We can begin by writing a matrix equation that is equivalent to the given system of equations. Recall that a system of linear equations ๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐,๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐,โฎโฎโฎโฑโฎโฎโฎ๐๐ฅ+๐๐ฅโฏ๐๐ฅ=๐.๏ง๏ง๏ง๏ง๏จ๏จ๏ง๏ ๏๏ง๏จ๏ง๏ง๏จ๏จ๏จ๏จ๏๏๏จ๏๏ง๏ง๏๏จ๏จ๏๏๏๏ is equivalent to the matrix equation โโโโ๐๐โฏ๐๐๐โฏ๐ โฎโฎโฑโฎ๐๐โฏ๐โโโโ โโโโ๐ฅ๐ฅโฎ๐ฅโโโโ =โโโโโ๐๐โฎ๐โโโโโ .๏ง๏ง๏ง๏จ๏ง๏๏จ๏ง๏จ๏จ๏จ ๏๏๏ง๏๏จ๏๏๏ง๏จ๏๏ง๏จ๏ The matrices in the equation above are called the coefficient, variable, and constant matrices respectively. From the given system of equations, our variables have names ๐ฅ, ๐ฆ, and ๐ง, which form the entries of the variable matrix. The constants 50โ โ120, 53โ โ560, and 55โ โ660 on the right-hand sides of the given equations form the entries of the constant matrix. Hence, the variable and constant matrices are, respectively, ๏ฟ๐ฅ๐ฆ๐ง๏,๏501205356055660๏. To find the coefficient matrix, we need to write down the coefficients of each variable in the correct order (that is, the order of ๐ฅ, ๐ฆ, and ๐ง) for each equation. This leads to the coefficient matrix ๏457415487419499410๏. Hence, the matrix equation is ๏4 57415487419499410๏๏ฟ๐ฅ๐ฆ๐ง๏=๏501205356055660๏. We can solve this equation by multiplying from the left the inverse of the coefficient matrix on both sides of equation above. Let us find the inverse of the coefficient matrix ๐ด=๏457415487419499410๏. We can use the adjoint method to obtain the inverse of this matrix, if it exists. Recall that a square matrix is invertible if its determinant is nonzero. Let us begin by computing the determinant of this matrix and making sure that it is nonzero. We recall that, for a 3ร3 matrix ๐ด=๏น๐๏ ๏๏ , its determinant can be computed by det๐ด=๐| ๐ด|โ๐|๐ด|+๐|๐ด|๏ง๏ง๏ง๏ง๏ง๏จ๏ง๏จ๏ง๏ฉ๏ง๏ฉ where ๐ด๏๏ are matrix minors obtained by taking the ๐th row and ๐th column from matrix ๐ด. We can apply this formula to our coefficient matrix ๐ด to obtain det๐ด=45||74199410||โ74||48194910||+45||48744994||=45ร(โ1046)โ74ร(โ451)+45ร886=โ406. Since det๐ดโ 0, we know that the inverse matrix ๐ด๏ฑ๏ง exists. Let us find the inverse. Recall that we can find the inverse matrix by using the adjoint method as follows:
Let us first find the cofactor matrix. Entries of the cofactor matrix are the determinants of the corresponding matrix minors multiplied by the alternating sign (โ1)๏๏ฐ๏ . We need to compute the determinants of 9 matrix minors with the corresponding sign for this purpose: +|๐ด|=+||74199410 ||=โ1046,โ|๐ด| =โ||48194910||=451,+|๐ด|=+||48744994||=886,โ|๐ด|=โ||74159410||=670,+ |๐ด|=+||451549 10||=โ285,โ|๐ด|=โ||45744994||=โ604,+|๐ด|=+||74157419||= 296,โ|๐ด|=โ||4 5154819||=โ135,+|๐ด|=+||45744874||=โ222.๏ง๏ง๏ง๏จ๏ง๏ฉ๏จ๏ง๏จ๏จ๏จ๏ฉ๏ฉ๏ง๏ฉ๏จ๏ฉ๏ฉ This leads to the cofactor matrix ๏โ 1046451886670โ285โ604296โ135โ222๏. We can find the adjoint matrix by taking the transpose: adj๐ด=๏โ1046670296451โ285โ 135886โ604โ22 2๏. Finally, by multiplying by the reciprocal of the determinant, which we computed earlier to be โ40 6, we obtain ๐ด=โ 1406๏โ1046670296451โ285โ135886โ604โ222๏.๏ฑ๏ง Recall that we can solve the matrix equation ๐ด๐=๐ต by writing ๐=๐ด๐ต๏ฑ๏ง. This leads to ๏ฟ๐ฅ๐ฆ๐ง ๏=โ1406๏โ1046670296451โ285โ135886โ604โ222๏๏5012053560 55660๏. Hence, we can finish by computing the matrix multiplication on the right-hand side of the equation above: ๏ฟ๐ฅ๐ฆ๐ง๏=โ1406๏โ 1046ร50120+670ร53560+296ร55 660451ร50120โ285ร5356โ135ร5 5660886ร50120โ604ร53560โ222ร55660๏=โ1406๏โ64960โ174580โ300440๏=๏160430740๏. This leads to ๐ฅ=160,๐ฆ=4 30,๐ง=740. Hence, the cost of a suite is 740 LE. Let us finish by recapping a few important concepts from this explainer. Key Points
How to solve a system of linear equations with an inverse matrix?Let A be the coefficient matrix, X be the variable matrix, and B be the constant matrix to solve a system of linear equations with an inverse matrix. As a result, weโd want to solve the system AX = B.
How do you find the inverse of a nonsingular matrix?Case 1: If A is a nonsingular matrix, it has an inverse. Let A be the coefficient matrix, X be the variable matrix, and B be the constant matrix to solve a system of linear equations with an inverse matrix. As a result, weโd want to solve the system AX=B.
How do you find the inverse of a system of equations?The method of determining the inverse is used to solve a system of linear equations, and it requires two additional matrices. The variables are represented by Matrix X. The constants are represented by Matrix B.
How do you solve AB = C with the inverse matrix?If A, B, and C are matrices in the matrix equation AB = C, and you want to solve for B, how do you do that? Just multiply by the inverse of matrix A (if the inverse exists), which you write like this: Now that you've simplified the basic equation, you need to calculate the inverse matrix in order to calculate the answer to the problem.
How do you solve equations using inverse operations?To use an inverse operation, just do the opposite of what the equation says! Use inverse operations to complete the equation. In this example 2 is being added to 7, to undo that operation we need to subtract by 2.
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