Matrices MCQ Questions Part I

This lesson is a set of multiple-choice questions on matrices for Class XII. It revises the order of a matrix and its number of elements, the general form written with indices, and each of the standard special matrices.

Order and number of elements

A matrix is a rectangular array of elements arranged in rows and columns. If a matrix has $m$ rows and $n$ columns, its order is $m \times n$, read as "$m$ by $n$".

The number of elements in such a matrix is the product of the rows and columns:

$\text{number of elements} = m \times n.$

If a matrix has $12$ elements, its possible orders are every factor pair of $12$:

$1 \times 12,\quad 2 \times 6,\quad 3 \times 4,\quad 4 \times 3,\quad 6 \times 2,\quad 12 \times 1.$

General form of a matrix

Writing a general element as $a_{ij}$ (row $i$, column $j$), a matrix of order $m \times n$ has the form

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}.$

Small cases. A $1 \times 1$ matrix is $\begin{bmatrix} a_{11} \end{bmatrix}$; a $1 \times 2$ matrix is $\begin{bmatrix} a_{11} & a_{12} \end{bmatrix}$; a $2 \times 2$ matrix is

$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}.$

A $2 \times 3$ matrix and a $3 \times 2$ matrix are

$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}, \qquad \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix},$

and a $3 \times 3$ matrix is

$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}.$

Row, column and square matrices

A matrix with a single row is a row matrix, for example $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$.

A matrix with a single column is a column matrix, for example

$\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix},$

which has order $3 \times 1$.

A matrix whose number of rows equals its number of columns is a square matrix.

Diagonal, scalar and identity matrices

In a diagonal matrix, every non-diagonal element is $0$, for example

$\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}, \qquad \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}.$

A scalar matrix is a diagonal matrix whose diagonal elements are all equal, for example

$\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}.$

An identity matrix is a diagonal matrix whose diagonal elements are all $1$:

$I_1 = \begin{bmatrix} 1 \end{bmatrix}, \qquad I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$

Zero matrix, negative of a matrix, and addition

In a zero matrix, every element is $0$, for example $\begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$. The zero matrix is denoted $O$, and it is the additive identity for matrices.

The negative $-A$ of a matrix $A$ is obtained by changing the sign of every element. For

$A = \begin{bmatrix} 1 & 2 \\ 0 & -1 \\ 0 & 4 \end{bmatrix}, \qquad -A = \begin{bmatrix} -1 & -2 \\ 0 & 1 \\ 0 & -4 \end{bmatrix}.$

To add or subtract two matrices, their orders must be equal.

Key takeaways

• A matrix of order $m \times n$ has $m \times n$ elements, and each factor pair of that count gives a possible order.
• The general element $a_{ij}$ sits in row $i$ and column $j$, which fixes the general form of any matrix.
• Diagonal, scalar and identity matrices are special square matrices; the zero matrix is the additive identity, and addition requires matrices of equal order.