Question

Which equation can be used to solve the matrix equation?

Answer 1

The correct option is 4.

Explanation:

The given matrix equation is

`\begin{bmatrix}(1)/(2) & (-1)/(4)\\ 2 & -(3)/(4)\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}2 & -4\\ 1 & 6\end{bmatrix}`

Let
`A=\begin{bmatrix}(1)/(2) & (-1)/(4)\\ 2 & -(3)/(4)\end{bmatrix}`,
`X=\begin{bmatrix}x\\ y\end{bmatrix}` and
`B=\begin{bmatrix}2 & -4\\ 1 & 6\end{bmatrix}`

The given equation can be written as

`AX=B`

If
`AX=B`, then
`X=A^(-1)B`

It means we have to find the matrix A⁻¹ .

`A=\begin{bmatrix}(1)/(2) & (-1)/(4)\\ 2 & -(3)/(4)\end{bmatrix}`

`|A|=(1)/(2)* (-3)/(4)-(-1)/(4)* 2=\frac{-3}[8}+(1)/(2)=(1)/(8)`

If
`P=\begin{bmatrix}a & b\\ c & d\end{bmatrix}`, then
`P^(-1)=(1)/(|P|)\begin{bmatrix}d &-b\\ -c & a\end{bmatrix}`

`A^(-1)=(1)/((1)/(8))\begin{bmatrix}-(3)/(4)& (1)/(4)\\ -2 &(1)/(2)\end{bmatrix}`

`A^(-1)=8\begin{bmatrix}-(3)/(4)& (1)/(4)\\ -2 &(1)/(2)\end{bmatrix}`

`A^(-1)=\begin{bmatrix}-6& 2\\ -16 &4\end{bmatrix}`

`X=A^(-1)B`

`\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}-6& 2\\ -16 &4\end{bmatrix}\begin{bmatrix}2 & -4\\ 1 & 6\end{bmatrix}`

Therefore the correct option is 4.