Name: Solving System of Linear Equations using Matrices
Uploaded: 2019-07-19
Duration: 11 min 30 s
Description: A system of linear equations can be represented in matrix form using its coefficient, variable and constant matrices. Learn about the inverse matrix method!

Question

Find the adjoint and inverse of the following matrix :

$A = (\begin{matrix} 3 & 2 & - 1 \\ 1 & 1 & 1 \\ 5 & 1 & - 1 \end{matrix})$

Accepted Answer

Given matrix is $A = (\begin{array}{r} 3 & 2 & - 1 \\ 1 & 1 & 1 \\ 5 & 1 & - 1 \end{array})$

We know that the adjoint of a square matrix is the transpose of its cofactor matrix.

Let $C_{i j}$ denotes the cofactor of $a_{i j}$ in $A = [\begin{matrix} a_{i j} \end{matrix}]$ . Then,

$C_{11} = (- 1)^{1 + 1} |\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}| = - 2$

$C_{12} = (- 1)^{1 + 2} |\begin{matrix} 1 & 1 \\ 5 & - 1 \end{matrix}| = 6$

$C_{13} = (- 1)^{1 + 3} |\begin{array}{l} 1 & 1 \\ 5 & 1 \end{array}| = - 4$

$C_{21} = (- 1)^{2 + 1} |\begin{array}{l} 2 & - 1 \\ 1 & - 1 \end{array}| = 1$

$C_{22} = (- 1)^{2 + 2} |\begin{array}{l} 3 & - 1 \\ 5 & - 1 \end{array}| = 2$

$C_{23} = (- 1)^{2 + 3} |\begin{array}{l} 3 & 2 \\ 5 & 1 \end{array}| = 7$

$C_{31} = (- 1)^{3 + 1} |\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}| = 3$

$C_{32} = (- 1)^{3 + 2} |\begin{matrix} 3 & - 1 \\ 1 & 1 \end{matrix}| = - 4$

$C_{33} = (- 1)^{3 + 3} |\begin{array}{l} 3 & 2 \\ 1 & 1 \end{array}| = 1$

Thus, the cofactor matrix of $A$ is $(\begin{matrix} - 2 & 6 & - 4 \\ 1 & 2 & 7 \\ 3 & - 4 & 1 \end{matrix})$

Therefore, $Adj (A) = {(\begin{matrix} - 2 & 6 & - 4 \\ 1 & 2 & 7 \\ 3 & - 4 & 1 \end{matrix})}^{T} =$ $(\begin{matrix} - 2 & 1 & 3 \\ 6 & 2 & - 4 \\ - 4 & 7 & 1 \end{matrix})$ .

We know that the inverse of $A$ is denoted by $A^{- 1}$ , and $A^{- 1} = \frac{1}{|A|} Adj (A)$ .

$|A| = |\begin{matrix} 3 & 2 & - 1 \\ 1 & 1 & 1 \\ 5 & 1 & - 1 \end{matrix}| = 3 (- 1 - 1) - 2 (- 1 - 5) - 1 (1 - 5)$

$= - 6 + 12 + 4 = 10$

Thus, $A^{- 1} = \frac{1}{10} (\begin{matrix} - 2 & 1 & 3 \\ 6 & 2 & - 4 \\ - 4 & 7 & 1 \end{matrix}) \begin{matrix} \end{matrix}$

Find the adjoint and inverse of the following matrix :
$A = (\begin{matrix} 3 & 2 & - 1 \\ 1 & 1 & 1 \\ 5 & 1 & - 1 \end{matrix})$

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Find the adjoint and inverse of the following matrix : A=32-111 151-1

Important Questions on Adjoint and Inverse of a Matrix

Learn and Prepare for any exam you want !

Find the adjoint and inverse of the following matrix :
$A = (\begin{matrix} 3 & 2 & - 1 \\ 1 & 1 & 1 \\ 5 & 1 & - 1 \end{matrix})$