Matrices and Determinants

A matrix is a rectangular array of real numbers enclosed in square brackets, arranged in horizontal rows and vertical columns. The order of a matrix is defined by $m \times n$, where $m$ represents the number of rows and $n$ the number of columns. Students must thoroughly understand various types of matrices, including row matrix, column matrix, rectangular matrix, square matrix, zero (null) matrix, identity matrix, and transpose of a matrix. Special attention should be given to symmetric and skew-symmetric matrices, as properties of transposes frequently appear in both short and long board questions.

Matrix operations form the computational core of this chapter. Addition and subtraction are only possible between matrices of the exact same order by adding or subtracting corresponding elements. Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix ($A_{m \times n}$ and $B_{n \times p}$). Students must remember that matrix multiplication is generally not commutative ($AB \neq BA$). Key algebraic laws, such as the associative and distributive laws, as well as the important transpose property $(AB)^T = B^T A^T$, are high-yield topics for board exams.

Every square matrix has a real number associated with it called its determinant, denoted by $|A|$ or $\det(A)$. A matrix with a determinant of zero is called a singular matrix, while a non-zero determinant makes it a non-singular matrix. The adjoint of a matrix is found by interchanging the diagonal elements and changing the signs of the off-diagonal elements. These concepts culminate in finding the multiplicative inverse of a matrix using the formula $A^{-1} = \frac{\text{adj}(A)}{|A|}$, which is essential for solving systems of linear equations using the matrix inversion method and Cramer's rule. Expect at least one long 8-mark question combining these methods in the final exam.