Recall that in order to solve a matrix equation we need to multiply both sides by the inverse of the coefficient matrix. Let's review the formula for calculating the inverse of a 2* 2 matrix.
A = a & b c & d A^(- 1) = 1/ad-bc d & - b - c & aNotice that for this formula to work we cannot have ad-bc=0. Furthermore, we can identify the quantity ad-bc. This is the determinant of matrix A.
|A| = |
cc
a & b
c & d
|
=
ad-bc
Even though we are talking about a 2* 2 matrix here, this condition applies to any square matrix. For a square matrix to have an inverse, its determinant has to be non-zero. Otherwise, we cannot solve the matrix equation this way. Therefore, we just need to find a matrix whose diagonal elements have the same product.
Example Solution [0.9em]
1cmA = 2 & 4 4 & 8 1cm [1.5em]
1cm|A| = (2)(8) - (4)(4) [0.9em]
|A| = 0 1cm
As the determinant of this matrix is zero, we cannot invert it to solve the matrix equation. The same would be true for any matrix with a null determinant; this just an example solution.
