Matrices — Part 2: Transpose, symmetric matrices and inverses: visual lesson, CBSE Class 12 Mathematics

Lesson

Matrices, Part 2: Transpose and inverse

Flip a matrix in its diagonal, and undo a matrix with its inverse.

In this lesson you will

Find the transpose of a matrix and use its four properties
Recognise symmetric and skew-symmetric matrices
Express a square matrix as the sum of a symmetric and a skew-symmetric matrix
Define an invertible matrix and prove that its inverse is unique
Find or verify the inverse of a 2 × 2 matrix using AB = BA = I

Rows of $A$ become columns of $A^T$.
$A$	$A^T$
$\begin{bmatrix} 1 & 4 & 0 \\ -2 & 3 & 5 \end{bmatrix}$	$\begin{bmatrix} 1 & -2 \\ 4 & 3 \\ 0 & 5 \end{bmatrix}$

Concept

What you will be able to do

Find the transpose $A^T$ and use its properties
Test for symmetric and skew-symmetric matrices
Split any square matrix into these two parts
Prove that an inverse is unique
Find or verify the inverse of a $2 \times 2$ matrix

Diagram

The transpose

Rows of $A$ become columns of $A^T$.
$A$	$A^T$
$\begin{bmatrix} 1 & 4 & 0 \\ -2 & 3 & 5 \end{bmatrix}$	$\begin{bmatrix} 1 & -2 \\ 4 & 3 \\ 0 & 5 \end{bmatrix}$

The transpose $A^T$ (also written $A'$ ) of an $m \times n$ matrix $A$ is the $n \times m$ matrix whose rows are the columns of $A$ : if $A = [a_{ij}]$ then $A^T = [a_{ji}]$ .

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