Question

The matrix equation represents a system of equations.

A matrix with 2 rows and 2 columns, where row 1 is 2 and 7 and row 2 is 2 and 6, is multiplied by matrix with 2 rows and 1 column, where row 1 is x and row 2 is y, equals a matrix with 2 rows and 1 column, where row 1 is 4 and row 2 is 6.

Solve for y using matrices. Show or explain all necessary steps.

Answer 1

y = -2

Explanation:

You want to solve this system of equations using matrices:

`\left[\begin{array}{cc}2&7\\2&6\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}4\\6\end{array}\right]`

Row operations

This system of equations lends itself to solution using a couple of row operations on the matrices involved.

r2 ⇒ r1 -r2

`\left[\begin{array}{cc}2&7\\0&1\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}4\\-2\end{array}\right]`

At this point, we know the value of y is -2.

The rest of the solution is obtained by ...

r1 ⇒ r1 -7·r2

`\left[\begin{array}{cc}2&0\\0&1\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}18\\-2\end{array}\right]`

r1 ⇒ r1/2

`\left[\begin{array}{cc}1&0\\0&1\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}9\\-2\end{array}\right]`

(x, y) = (9, -2)

Inverse matrix

The inverse of the coefficient matrix is the transpose of the cofactor matrix, divided by the determinant. The determinant is 2·6 -2·7 = -2. Then the inverse matrix is ...

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

Multiplying the original equation by this gives ...

`\left[\begin{array}{cc}-3&7/2\\1&-1\end{array}\right] \left[\begin{array}{cc}2&7\\2&6\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{cc}-3&7/2\\1&-1\end{array}\right]\left[\begin{array}{c}4\\6\end{array}\right]\\\\\\\left[\begin{array}{cc}1&0\\0&1\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right] =\left[\begin{array}{c}-3\cdot4+(7)/(2)\cdot6\\1\cdot4-1\cdot6\end{array}\right]`

So, finally, ...

`\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{c}9\\-2\end{array}\right]`

The value of y is -2.