Castle Learning Reference

Reference Tables for Pre-Calculus

Matrices

Matrix Dimensions

A matrix is a table of numbers used to represent data. The dimensions of a matrix are the number of rows × the number of columns. Each entry in a matrix can be referred to by its row and column number.

For example, A =

is a 3 × 2 matrix.

A =

is an m × n matrix.

Adding and Subtracting Matrices

Matrices with the same dimensions can be added and subtracted. Just add or subtract the corresponding entries.

Multiplying Matrices

For two matrices to be multiplied, the two inner dimensions must match. That is, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting product matrix is has the same number of rows as the first matrix and the same number of columns as the second matrix. For example, the product of a 3 × 2 matrix and a 2 × 5 matrix would be a 3 × 5 matrix.

Here are some more examples of the dimensions of matrices that can be multiplied and dimensions of the resulting matrix.
(1 × 3) · (3 × 4) = (1 × 4)
(2 × 5) · (5 × 3) = (2 × 3)
(2 × 3) · (4 × 1): This product cannot be formed because the inner dimensions do not match.

Here is the rule for multiplying 2 × 2 matrices.

Here is an example of how to multiply a (2 × 2) matrix by a (2 × 4) matrix. The resulting matrix will have the dimensions of (2 × 4).

Here is an example of how to multiply a (3 × 1) matrix by a (1 × 2) matrix. The resulting matrix will have the dimensions of (3 × 2).

Determinant of a Matrix

For the 2 × 2 matrix

, the determinant is ad – bc.
If the determinant = 0, the matrix does not have an inverse.

Inverse of a Matrix

The product of a matrix and its inverse is the identity matrix.

Only a square matrix (n × n) can have an inverse.
If the determinant = 0, the matrix does not have an inverse.

For the 2 × 2 matrix

, the inverse matrix is