Matrices — common mistakes

In exams, if you are asked to find the inverse of a 3x3 matrix, you MUST show your method (determinant, cofactors, adjugate). Do not just write down the answer from your calculator. Use your calculator to check your final answer. A quick check is to multiply your calculated inverse by the original matrix; you should get the identity matrix.

The process of matrix multiplication (row-by-column) is order-dependent. The element in row 1, column 1 of AB is (row 1 of A) • (column 1 of B), while for BA it's (row 1 of B) • (column 1 of A). These involve different numbers and will generally produce different results. For example, if $\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, then $\mathbf{AB} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$ but $\mathbf{BA} = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$.

If $\det(\mathbf{A}) = 0$, the matrix inverse method fails because $\mathbf{A}^{-1}$ doesn't exist. This means the system does not have a unique solution. It will either have no solutions (inconsistent system) or infinitely many solutions (dependent system). To determine which, you would need to use other methods, such as Gaussian elimination, and analyse the geometric interpretation (e.g., parallel planes, planes intersecting in a line).

No. Exam questions are designed to test your understanding of the process. Marks are awarded for showing the correct method: calculating the determinant, finding the matrix of cofactors, forming the adjugate matrix, and then stating the inverse. You should absolutely use your calculator to check your final answer, but you cannot skip the working.