Matrices

Understand that a matrix is a rectangular array of numbers, symbols, or expressions.
Be able to identify the order of a matrix. The order of a matrix is represented as m×n where ‘m’ is the number of rows and ‘n’ is the number of columns.
Learn how to perform matrix operations such as addition, subtraction, multiplication, scalar multiplication and division.
Remember that matrix multiplication is not commutative (i.e., AB ≠ BA), but matrix addition is commutative (i.e., A+B = B+A).
Grasp the concepts of zero (or null) matrix, unit (or identity) matrix and diagonal matrix.
Understand that the scalar multiplication is associative (i.e., a(bc) = (ab)c).
Be aware that for any matrix, adding zero matrix does not change the original matrix (i.e., A + 0 = A).
Remember that the subtraction of two matrices A and B is equivalent to adding A and -B. Thus the operation can be written as A – B = A + (-B).
Realise that the determinant is a special number that can only be calculated for square matrices (2x2, 3x3, etc.). The determinant helps us find the inverse of a matrix and tells us things about the matrix that are useful in systems of linear equations and calculus.
Be capable of using Cramer’s Rule to solve systems of two and three linear equations.
Understand how to calculate the inverse of a matrix if it exists. Note that not all matrices have inverses. A matrix must be square (i.e., have the same number of rows as columns) and its determinant does not equal to zero.
Be able to use the adjoint or adjugate method to calculate the inverse of a matrix.
Remember that the power of a matrix (A^n where n is a non-negative integer) is not the same as raising each entry to the power.
Comprehend the concept of eigenvalues and eigenvectors and their applications in different areas of mathematics.
Understand the transformations that can be achieved with matrices in two and three dimensions, including rotation, reflection, shearing, and scaling.
Gain skills to perform row operations and use Gaussian or Gauss-Jordan elimination to reduce matrices to echelon form or reduced echelon form.
Be aware of properties and applications of special types of matrices such as orthogonal matrices and symmetric matrices.
Lastly, practice solving problems and interpreting results related to matrices. This includes understanding real-world applications of matrices in various fields such as physics, computer graphics, and economics.