SPM Mathematics: How to Master Matrices

Adding and subtracting matrices only works when they are the same size, and multiplication follows the row-by-column rule that trips students who rush. Write each entry in its own grid position rather than crowding the working.

Multiplication is not commutative, so AB and BA can differ; keep the order the question gives you.

The inverse of a 2x2 matrix comes from a fixed recipe: swap the leading diagonal, negate the other two entries, and divide by the determinant. If the determinant is zero there is no inverse, which is itself a valid answer.

Because the steps never change, this is one of the most dependable ways to secure marks, so practise it until it is automatic.

Two simultaneous linear equations can be written as a matrix equation and solved with the inverse, a method the exam specifically asks for. Set up the coefficient matrix carefully, since one misplaced sign spoils the whole solution.

Then check your answer by substituting back into the original equations, which costs seconds and catches slips.

Matrix addition and subtraction are the gentle part of this topic: two matrices can only be added or subtracted if they have exactly the same order, and the operation is done entry by entry in matching positions. Multiplication is where the rules matter far more.

Two matrices can only be multiplied if the number of columns in the first matches the number of rows in the second, this is worth checking before writing a single number, because it tells you immediately whether the multiplication is even possible and what order the answer matrix will have. The calculation itself follows a "row times column" pattern: each entry in the answer comes from multiplying corresponding entries of a row from the first matrix with a column from the second matrix, then adding those products together.

Working through this slowly, row by row, with the matrices laid out clearly on paper rather than attempted mentally, is the difference between a clean full-marks answer and a small slip that costs the whole question.

After using the inverse of a matrix to solve a pair of simultaneous equations, it's worth spending thirty seconds substituting your values of x and y back into both original equations before moving on. Because the method involves several steps, finding the determinant, forming the inverse, then multiplying two matrices together, there are several places a small arithmetic slip can happen without the working looking obviously wrong.

Substituting back catches this reliably: if both original equations balance with your values, the answer is almost certainly correct; if either equation doesn't balance, you know to recheck your matrix multiplication before submitting the answer. This check takes far less time than reworking the whole problem from scratch, and it's especially worth doing in Paper 2, where a wrong final answer built on a correct method can still cost the accuracy mark even though the method marks are awarded.

Get into the habit of treating this substitution step as part of finishing the question, not as an optional extra only done if time remains.