Answer to Question #118524 in Linear Algebra for Jflows

Question

Consider A\\ =\\left[\\begin{array}{cc} 2 & 3\\\\ 1 & 2\\end{array}\\right],\\ B\\ =\\left[\\begin{array}{cc} 1 & 5\\\\ \\end{array}\\right]; what is the identity element of the two matrices
a.\\(\\left[\\begin{array}{cc} 0 & 1\\\\ 1 & 0 \\end{array}\\right]\\)
b.\\(\\left[\\begin{array}{cc} 1 & 0\\\\ 0 & 1 \\end{array}\\right]\\)
c.\\(\\left[\\begin{array}{cc} 0 & 0\\\\ 1 & 1 \\end{array}\\right]\\)
d.\\(\\left[\\begin{array}{cc} 2 & -2\\\\ -13 \\end{array}\\right]\\)

Answer 1

Given $A = \begin{bmatrix} 2 & 3 \\ 1 & 2\end{bmatrix}, B = \begin{bmatrix} 1 & 5 \end{bmatrix}$ .

Let $I$ be the identity then $A I = A, BI = B$.

Assume $I = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$ then

$AI = A \implies \begin{bmatrix} 2a+3c & 2b+3d \\ a+2c & b+2d\end{bmatrix} = \begin{bmatrix} 2& 3 \\ 1 & 2 \end{bmatrix}$

So, $2a+3c = 2, 2b+3d = 3, a+2c = 1, b+2d = 2.$

Also, $BI = B \implies \begin{bmatrix} a+5c & b+5d \end{bmatrix} = \begin{bmatrix} 1 & 5 \end{bmatrix}$

$\implies a+5c = 1, b+5d = 5.$

Hence, $3d=3 \implies d=1$

and $7c = 0 \implies c = 0$

Hence, $a = 1, b = 0$

Hence, $I = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ .

Option (b) is the correct option.