What is the solution of the matrix equation?

Question

[9 4];[2 1] X= [-9 -6];[-1 -8]

Answer 1

To find the solution of the matrix equation, we need to find the inverse of the coefficient matrix and multiply it by the right-hand side matrix.

The coefficient matrix is given by:

[9 4]
[2 1]

To calculate the inverse, we can use the formula for a 2x2 matrix:

[1/det(A)] * [d -b]
[-c a]

where det(A) is the determinant of matrix A, a is the upper left element, b is the upper right element, c is the lower left element, and d is the lower right element.

In this case, a = 9, b = 4, c = 2, and d = 1. Therefore, we have:

det(A) = (9*1) - (4*2) = 9 - 8 = 1

Now, we can calculate the inverse:

[1/det(A)] * [d -b]
[-c a] = [1/1] * [1 -4]
[-2 9] = [1 -4]
[-2 9]

The inverse matrix is:

[1 -4]
[-2 9]

Now, we can multiply the inverse matrix by the right-hand side matrix:

[1 -4] [-9 -6] [(-9*1) + (-6*-2) (-9*-4) + (-6*9)]
[-2 9] * [-1 -8] = [(-2*1) + (9*-2) (-2*-4) + (9*9)]

Simplifying the calculations, we have:

[1 -4] [-9 -6] [-9 + 12 36 - 54]
[-2 9] * [-1 -8] = [-2 -18 -8 + 81]

[1 -4] [-9 -6] [3 -18]
[-2 9] * [-1 -8] = [-20 73]

Therefore, the solution to the matrix equation is:

X = [3 -18]
[-20 73]