Name: The Applications of Matrices
Uploaded: 2019-07-19
Duration: 7 min 47 s
Description: Let's learn the applications of matrices, such as solving homogenous and non-homogenous linear equations, the Gauss-Jordan method, and the reduction method.

Question

Find the inverse of

[\begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{matrix}]

by adjoint method.

Accepted Answer

Let $A = [\begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{matrix}]$

$Co - factor matrix = [\begin{matrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{matrix}]$

Co-factors of matrix $A$ are

$C_{11} = + |\begin{matrix} 2 & 3 \\ 2 & 1 \end{matrix}| = (2 - 6) = - 4, C_{12} = - |\begin{matrix} 0 & 3 \\ 1 & 1 \end{matrix}| = - (0 - 3) = 3$

$C_{13} = + |\begin{matrix} 0 & 2 \\ 1 & 2 \end{matrix}| = (0 - 2) = - 2, C_{21} = - |\begin{matrix} 0 & 1 \\ 2 & 1 \end{matrix}| = - (0 - 2) = 2$

$C_{22} = + |\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}| = (1 - 1) = 0, C_{23} = - |\begin{matrix} 1 & 0 \\ 1 & 2 \end{matrix}| = - (2 - 0) = - 2$

$C_{31} = + |\begin{matrix} 0 & 1 \\ 2 & 3 \end{matrix}| = (0 - 2) = - 2, C_{32} = - |\begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}| = - (3 - 0) = - 3$

$C_{33} = + |\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}| = (2 - 0) = 2$

Therefore, co-factor matrix of $A$ is $[\begin{matrix} - 4 & 3 & - 2 \\ 2 & 0 & - 2 \\ - 2 & - 3 & 2 \end{matrix}]$

Adjoint matrix of $A$ is ${[\begin{matrix} - 4 & 3 & - 2 \\ 2 & 0 & - 2 \\ - 2 & - 3 & 2 \end{matrix}]}^{T} = [\begin{matrix} - 4 & 2 & - 2 \\ 3 & 0 & - 3 \\ - 2 & - 2 & 2 \end{matrix}]$

Now, $|A| = 1 |\begin{matrix} 2 & 3 \\ 2 & 1 \end{matrix}| - 0 |\begin{matrix} 0 & 3 \\ 1 & 1 \end{matrix}| + 1 |\begin{matrix} 0 & 2 \\ 1 & 2 \end{matrix}|$

$|A| = 1 (2 - 6) - 0 + 1 (0 - 2)$

$|A| = - 4 - 2 = - 6$

Now, $A^{- 1} = \frac{1}{|A|} [\begin{matrix} - 4 & 2 & - 2 \\ 3 & 0 & - 3 \\ - 2 & - 2 & 2 \end{matrix}]$

$A^{- 1} = \frac{1}{- 6} [\begin{matrix} - 4 & 2 & - 2 \\ 3 & 0 & - 3 \\ - 2 & - 2 & 2 \end{matrix}]$

Therefore, $A^{- 1} = [\begin{matrix} \frac{2}{3} & - \frac{1}{3} & \frac{1}{3} \\ - \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{3} & - \frac{1}{3} \end{matrix}]$

Find the inverse of $[\begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{matrix}]$ by adjoint method.

Important Questions on Matrices

Learn and Prepare for any exam you want !

Find the inverse of 101023121 by adjoint method.

Important Questions on Matrices

Learn and Prepare for any exam you want !

Find the inverse of $[\begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{matrix}]$ by adjoint method.