Question

The matrix equation represents a system of equations.

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

Answer 1

(x, y) = (-4, 3)

Explanation:

You want the solution to the matrix equation ...

`\left[\begin{array}{cc}2&5\\1&3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] =\left[\begin{array}{c}7\\5\end{array}\right]`

Row Reduction

Perhaps the easiest solution is to let a calculator reduce the augmented matrix to row-echelon form. This is done in the attachment. It shows the solution is (x, y) = (-4, 3).

Inverse matrix

Alternatively, you can left-multiply the equation by the inverse of the coefficient matrix. The inverse of the 2×2 coefficient matrix is easily found.

The inverse is the transpose of the cofactor matrix, divided by the determinant. For a 2×2 matrix, the transpose of the cofactor matrix negates the off-diagonal terms and swaps the diagonal terms. The determinant is the difference of the products of the diagonal and off-diagonal terms:

`\left[\begin{array}{cc}2&5\\1&3\end{array}\right]^(-1)=(1)/(2\cdot3-1\cdot5)\left[\begin{array}{cc}3&-5\\-1&2\end{array}\right] =\left[\begin{array}{cc}3&-5\\-1&2\end{array}\right]`

Multiplying this by the constant vector, we have ...

x = 3(7) -5(5) = -4

y = -1(7) +2(5) = 3

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Additional comment

Using row-reduction avoids the problem of the inverse matrix being undefined — when the determinant is zero. That happens when the equations are dependent or inconsistent. The result of row-reduction will tell you which of those is the case: the bottom row is 0 for dependent equations, and 0 0 1 for inconsistent equations.