The inverse of begin{bmatrix} 1 & 2 & 3 0 & | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{bmatrix}$.First,we calculate the determinant $|A| = 1(1 	imes 1 - 2 	imes 0) -

Vedclass · Accepted Answer

$\begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix}$

Vedclass · Answer

(B) Let $A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{bmatrix}$.First,we calculate the determinant $|A| = 1(1 	imes 1 - 2 	imes 0) - 2(0 	imes 1 - 2 	imes 0) + 3(0 	imes 0 - 1 	imes 0) = 1(1) = 1$.Since $|A| 
eq 0$,the inverse $A^{-1}$ exists.Next,we find the matrix of cofactors $C_{ij}$:$C_{11} = +|\begin{smallmatrix} 1 & 2 \ 0 & 1 \end{smallmatrix}| = 1, C_{12} = -|\begin{smallmatrix} 0 & 2 \ 0 & 1 \end{smallmatrix}| = 0, C_{13} = +|\begin{smallmatrix} 0 & 1 \ 0 & 0 \end{smallmatrix}| = 0$$C_{21} = -|\begin{smallmatrix} 2 & 3 \ 0 & 1 \end{smallmatrix}| = -2, C_{22} = +|\begin{smallmatrix} 1 & 3 \ 0 & 1 \end{smallmatrix}| = 1, C_{23} = -|\begin{smallmatrix} 1 & 2 \ 0 & 0 \end{smallmatrix}| = 0$$C_{31} = +|\begin{smallmatrix} 2 & 3 \ 1 & 2 \end{smallmatrix}| = 4-3 = 1, C_{32} = -|\begin{smallmatrix} 1 & 3 \ 0 & 2 \end{smallmatrix}| = -2, C_{33} = +|\begin{smallmatrix} 1 & 2 \ 0 & 1 \end{smallmatrix}| = 1$Thus,$Adj(A) = \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix}$.Finally,$A^{-1} = \frac{1}{|A|} Adj(A) = \frac{1}{1} \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix}$.

The inverse of $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$ is

Similar Questions

If $P = \begin{vmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{vmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $\det(A) = 4$, then $\alpha$ is equal to

Find the inverse of the matrix $\left[\begin{array}{cc}-1 & 5 \\ -3 & 2\end{array}\right]$ (if it exists).

If $a, b, c$ and $d$ are real numbers such that $a^2+b^2+c^2+d^2=1$ and $A=\left[\begin{array}{cc}a+ib & c+id \\ -c+id & a-ib\end{array}\right]$, then $A^{-1}$ is equal to

If matrices $X$ and $Y$ are inverses of each other,then which of the following is true?

If $A$ and $B$ are non-singular matrices of order $2$ such that $(AB)^{-1} = \frac{1}{6} \begin{bmatrix} -7 & -3 \\ 2 & 3 \end{bmatrix}$ and $A^{-1} = \frac{1}{3} \begin{bmatrix} 4 & 3 \\ -1 & 0 \end{bmatrix}$,then $B^{-1} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module