Matrices · Mathematics · Class 12

If $A=\left[\begin{array}{ccc}a & c & -1 \\ b & 0 & 5 \\ 1 & -5 & 0\end{array}\right]$ is a skew-symmetric matrix, then the value of $2 a-(b+c)$ is

If $\left[\begin{array}{lll}x & 2 & 0\end{array}\right]\left[\begin{array}{c}5 \\ -1 \\ x\end{array}\right]=\left[\begin{array}{ll}3 & 1\end{array}\right]\left[\begin{array}{c}-2 \\ x\end{array}\right]$, then value of $x$ is

Find the matrix $\mathrm{A}^2$, where $A=\left[a_{i j}\right]$ is a $2 \times 2$ matrix whose elements are given by $a_{i j}=$ maximum $(i, j)-$ minimum $(i, j)$

If for a square matrix $$\mathrm{A}, A^2-A+I=\mathrm{O}$$, then $$\mathrm{A}^{-1}$$ equals:

$$\text { If } A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right], B=\left[\begin{array}{ll} x & 0 \\ 1 & 1 \end{array}\right] \text { and } A=B^2 \text {, then } x \text { equals: }$$

(a) If $A=\left[\begin{array}{ccc}1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0\end{array}\right]$, then find $A^{-1}$ and hence solve the following system of equations:

$$\begin{array}{r} x+2 y-3 z=1 \\ 2 x-3 z=2 \\ x+2 y=3 \end{array}$$

OR

(b) Find the product of the matrices $\left[\begin{array}{ccc}1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4\end{array}\right]\left[\begin{array}{ccc}-6 & 17 & 13 \\ 14 & 5 & -8 \\ -15 & 9 & -1\end{array}\right]$ and hence solve the system of linear equations:

$$\begin{aligned} x+2 y-3 z & =4 \\ 2 x+3 y+2 z & =2 \\ 3 x-3 y-4 z & =11 \end{aligned}$$

$$\text { If } A=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{array}\right] \text {, then show that } A^3-23 A-40 I=O \text {. }$$