Browse Questions

# If Matrix A = $\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$, and $A^2=kA$, then write the value of $k$.

Toolbox:
• The scalar multiplication $cA$ of a matrix $A$ and a number $c$ (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of $A$ by $c$.
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Given $\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$, L.H.S. $A^2 = \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$
$\Rightarrow$ L.H.S. $A^2 = \begin{bmatrix} 2 &-2 \\ -2 & 2 \end{bmatrix} = 2 \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$
R.H.S. $= k A = k \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$
Since L.H.S. = R.H.S., we can see that $2 \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} = k \begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} \rightarrow k = 2$