Mathematics

Matrices and Matrix Algebra: Quick Revision

A brief note on matrices and matrix algebra: the order and types of matrix, sums and scalar multiples, products and why the order matters, the transpose, and symmetric and skew-symmetric matrices. With a worked example, practice, a program and questions to check yourself.

Written by StemPy Lab

A brief note on Matrices and Matrix Algebra. Read it before a test. Try each question at the end before you open its answer.

Matrices and their types

Order: a matrix with $m$ rows and $n$ columns has order $m \times n$ . The entry $a_{ij}$ is in row $i$ , column $j$ .
Equal matrices: the same order and equal entries in every place.
Special matrices: the zero matrix $O$ , the identity $I$ , and scalar, diagonal and triangular matrices.
Sum and scalar multiple: add entry by entry (same order only). $kA$ multiplies every entry.

Products

Exists when: the columns of $A$ equal the rows of $B$ . An $m \times n$ matrix times an $n \times p$ matrix is $m \times p$ .
The rule: $(AB)_{ij}$ is row $i$ of $A$ times column $j$ of $B$ .
Kept: $(AB)C = A(BC)$ , $A(B + C) = AB + AC$ , $AI = IA = A$ .
Lost: $AB \ne BA$ in general. $AB = O$ does not force $A = O$ or $B = O$ . $(A + B)^2 = A^2 + AB + BA + B^2$ .

The transpose

Rules: $(A^T)^T = A$ , $(A + B)^T = A^T + B^T$ , $(kA)^T = kA^T$ , $(AB)^T = B^TA^T$ .
Symmetric: $A^T = A$ . Skew-symmetric: $A^T = -A$ , with a zero diagonal.
The split: $A = \tfrac{1}{2}(A + A^T) + \tfrac{1}{2}(A - A^T)$ , a symmetric part plus a skew-symmetric part, in one way only.

A worked example

For $A =

\displaystyle \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

$and$ B =

\displaystyle \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix}

$, is$ (AB)^T = A^TB^T$?

Find $AB$ . $AB =

\displaystyle \begin{bmatrix} 4 & 2 \\ 10 & 4 \end{bmatrix}

$, so$ (AB)^T =

\displaystyle \begin{bmatrix} 4 & 10 \\ 2 & 4 \end{bmatrix}

Find $A^TB^T$ . $A^T =

\displaystyle \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

$and$ B^T =

\displaystyle \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}

$, so$ A^TB^T =

\displaystyle \begin{bmatrix} 2 & 4 \\ 4 & 6 \end{bmatrix}

Compare. They differ. The correct rule is $(AB)^T = B^TA^T$ .

Practice with new numbers

A program to try

The program checks three rules of this note on two matrices: addition commutes, multiplication does not, and the transpose of a product reverses the order. It also splits a matrix into its symmetric and skew parts.

Mistakes to avoid

Multiplying place by place. A product is row by column.
Assuming $AB = BA$ . Check the order of the factors every time.
Using $(A + B)^2 = A^2 + 2AB + B^2$ . It needs $AB = BA$ .
Writing $(AB)^T = A^TB^T$ . The order reverses.
A non-zero diagonal in a skew-symmetric matrix. It must be all zeros.

Check yourself

$A$ is $3 \times 2$ and $B$ is $2 \times 5$ . What is the order of $AB$ ?

Find $

\displaystyle \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}

\displaystyle \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}

\displaystyle \begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}

Is $A + A^T$ symmetric or skew-symmetric?

Find the skew-symmetric part of $

\displaystyle \begin{bmatrix} 2 & 6 \\ 2 & 5 \end{bmatrix}

\displaystyle \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}

If $AB = O$ , must $A = O$ or $B = O$ ?

\displaystyle \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}

$and$

\displaystyle \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}

Sources

NCERT, Mathematics Textbook for Class XII, Part I, Chapter 3 (Matrices).
OpenStax, Precalculus 2e, Section 9.5 (Matrices and Matrix Operations).

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