Question

In the matrix equation below, what are the values of x and y?

Answer 1

The values of x and y are 5 and 2 respectively.

Explanation:

We have given a matrix equation.

`(1)/(2) \left[\begin{array}{ccc}4&8\\x+3&-4\\\end{array}\right] -3\left[\begin{array}{ccc}1&y+1\\-1&-2\end{array}\right] =\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]`

We have to find the value of x and y.

Multiplying the scalars into the matrix, we have

`\left[\begin{array}{ccc}2&4\\(x+3)/(2)&-2\end{array}\right] -\left[\begin{array}{ccc}3&3y+3\\-3&-6\end{array}\right] =\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]`

Adding the matrices of left sides, we have

`\left[\begin{array}{ccc}2-3&4-3y-3\\(x+3)/(2)+3&-2+6\end{array}\right]` =
`\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]`

`\left[\begin{array}{ccc}-1&1-3y\\(x+9)/(2)&4\end{array}\right]` =
`\left[\begin{array}{ccc}-1&-5\\7&4\end{array}\right]`

Two matrices are equal if their entries are equal.

1-3y = -5 ⇒ -3y = -5-1 ⇒ -3y = -6

y = 2

`(x+9)/(2) =7`

x+9 = 14

x = 14-9

x = 5

Hence, the values of x and y are 5 and 2 respectively.

Answer 2

`x=5,y=2`

Explanation:

The given matrix equation is

`(1)/(2) \left[\begin{array}{cc}4&8\\x+3&-4\end{array}\right] -3\left[\begin{array}{cc}1&y+1\\-1&-2\end{array}\right]=\left[\begin{array}{cc}-1&-5\\7&4\end{array}\right]`

We multiply the scalars to get;

`\left[\begin{array}{cc}2&4\\ (x+3)/(2)&-2\end{array}\right] --\left[\begin{array}{cc}3&3y+3\\-3&-6\end{array}\right]=\left[\begin{array}{cc}-1&-5\\7&4\end{array}\right]`

We simplify the right hand side to get;

`\left[\begin{array}{cc}-1&1-3y\\ (x+3)/(2)+3&4\end{array}\right] =\left[\begin{array}{cc}-1&-5\\7&4\end{array}\right]`

The two matrices are equal. This means the corresponding entries are also equal.

`\Rightarrow 1-3y=-5`

`\Rightarrow -3y=-5-1`

`\Rightarrow -3y=-6`

`\Rightarrow y=2`

Also

`(x+3)/(2)+3=7`

`\Rightarrow (x+3)/(2)=7-3`

`\Rightarrow (x+3)/(2)=4`

`\Rightarrow x+3=8`

`\Rightarrow x=8-3`

`\Rightarrow x=5`