### Types of Matrices

• Identify types of matrices such as Square, Diagonal, Scalar, Row, Column, Identity.

## Row Matrix

A matrix which has only one row.

$\mathrm{Example\ of\ Row\ Matrix} \large \begin{bmatrix} 2 & -6& 4 \end{bmatrix}$

## Column Matrix:

A matrix which has only one column.

$\mathrm{Example\ of\ Column\ Matrix} \large \begin{bmatrix} 3\\ 4\\ -1 \end{bmatrix}$

## Square Matrix:

A matrix having same no. of rows and columns.

$\mathrm{Example\ of\ Square\ Matrix} \large \begin{bmatrix} 3 & 6 & 1\\ 4&5 & 2\\ 2& 0 & -2 \end{bmatrix}$

## Diagonal Matrix:

A square matrix where only diagonal elements are present, non-diagonal elements are zero.

$\mathrm{Example\ of\ Diagonal\ Matrix} \large \begin{bmatrix} 3 & 0 & 0\\ 0&5 & 0\\ 0& 0 & -2 \end{bmatrix}$

## Scalar Matrix:

Diagonal matrix where all diagonal elements are identical.

$\mathrm{Example\ of\ Scalar\ Matrix} \large \begin{bmatrix} 3 & 0 & 0\\ 0&3 & 0\\ 0& 0 & 3 \end{bmatrix}$

## Identity Matrix:

Scalar Matrix where the diagonal elements are one.

$\mathrm{Example\ of\ Identity\ Matrix} \large \begin{bmatrix} 1 & 0 & 0\\ 0&1 & 0\\ 0& 0 & 1 \end{bmatrix}$

## Singular Matrix:

A Matrix whose determinant is equal to zero.

$\mathrm{Example\ of\ Singular\ Matrix} \large \begin{bmatrix} 2 & 3\\ -6 & -9 \end{bmatrix}$

### Solved Example:

#### 4-1-01

For which value of x will the matrix given below become singular? $\begin{bmatrix} 8 & x & 0\\ 4 & 0 & 2\\ 12 & 6 & 0 \end{bmatrix}$

Solution:
For singularity of the matrix, |A| = 0 $\begin{vmatrix} 8 & x & 0 \\ 4 & 0 & 2\\ 12 & 6 & 0 \end{vmatrix} = 0$ \begin{align*} 8[0 - 2 \times 6] - x[0 - 24] + 0[24 - 0] &= 0\\ 8 \times(- 12) + 24x &= 0\\ -96 + 24x &= 0 \\ x &= \dfrac{96}{24} = 4 \end{align*}

### Solved Example:

#### 4-1-02

If A is a square matrix then orthogonality property mandates: (GATE Civil 2021)

Solution:
If $AA^T = I$ or $A^{-1} = A^T$, the matrix is called Orthogonal.