46  Matrices

46.1 Introduction

The concept of matrices provides a compact way to represent and manipulate data, equations, and transformations.
In aptitude and higher mathematics, matrices are used for solving systems of equations, transformations in geometry, computer graphics, statistics, economics, and more.

This chapter covers definitions, types of matrices, operations, properties, determinants (briefly), and exam-style problems.

46.2 Definition of a Matrix

A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns.

General form of an \(m \times n\) matrix:

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

  • \(m\) = number of rows
  • \(n\) = number of columns
  • \(a_{ij}\) = element in \(i\)-th row, \(j\)-th column

46.3 Types of Matrices

  • Row matrix: only one row.
  • Column matrix: only one column.
  • Square matrix: number of rows = number of columns (\(n \times n\)).
  • Zero matrix: all entries 0.
  • Diagonal matrix: all non-diagonal entries 0.
  • Scalar matrix: diagonal matrix with all diagonal elements equal.
  • Identity matrix (\(I_n\)): diagonal elements = 1.
  • Upper/Lower triangular matrix: all entries below/above diagonal 0.
  • Symmetric matrix: \(A^T = A\).
  • Skew-symmetric matrix: \(A^T = -A\).

46.4 Equality of Matrices

Two matrices \(A\) and \(B\) are equal if: 1. They have the same order.
2. All corresponding elements are equal: \(a_{ij} = b_{ij}\).

46.5 Matrix Operations

46.5.1 Addition & Subtraction

Defined only for matrices of same order: \[ A+B=[a_{ij}+b_{ij}],\;\; A-B=[a_{ij}-b_{ij}] \]

Properties:
- Commutative: \(A+B=B+A\)
- Associative: \((A+B)+C=A+(B+C)\)

46.5.2 Scalar Multiplication

Each element multiplied by scalar \(k\):
\[ kA=[ka_{ij}] \]

46.5.3 Matrix Multiplication

If \(A\) is \(m \times n\) and \(B\) is \(n \times p\), product \(AB\) is \(m \times p\) with: \[ (AB)_{ij}=\sum_{k=1}^n a_{ik}b_{kj} \]

Key notes:
- Order matters: \(AB \neq BA\) in general.
- Defined only when inner dimensions match.

46.5.4 Properties

  • Associative: \(A(BC)=(AB)C\)
  • Distributive: \(A(B+C)=AB+AC\)
  • Not commutative generally: \(AB\neq BA\)

46.6 Transpose of a Matrix

If \(A=[a_{ij}]_{m\times n}\), then \(A^T=[a_{ji}]_{n\times m}\).

Properties:
- \((A^T)^T=A\)
- \((A+B)^T=A^T+B^T\)
- \((kA)^T=kA^T\)
- \((AB)^T=B^T A^T\)

46.7 Determinant of a Square Matrix (Brief Recall)

For a \(2\times2\) matrix:
\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc \]

For \(3\times3\): expansion by minors and cofactors.

46.8 Inverse of a Matrix

For a non-singular square matrix \(A\): \[ A^{-1} = \frac{1}{|A|}\text{Adj}(A) \] where \(|A|\neq 0\).

Property: \(AA^{-1}=I\).

46.9 Rank of a Matrix (Intro)

Rank = maximum number of linearly independent rows/columns.
Used in solving equations. (Advanced: not usually asked in IPMAT but useful to know.)

46.10 System of Linear Equations (Matrix Form)

System: \[ a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \]

can be written as \(AX=B\), where:
- \(A\) = coefficient matrix
- \(X\) = variable column
- \(B\) = constants column

Solution often uses inverse: \(X=A^{-1}B\) (if \(|A|\neq 0\)).

46.11 Solved Examples

Example 1
If \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\), \(B=\begin{bmatrix}4&3\\2&1\end{bmatrix}\), find \(A+B\).

\(A+B=\begin{bmatrix}5&5\\5&5\end{bmatrix}\)

Example 2
Find determinant of \(\begin{bmatrix}2&3\\5&7\end{bmatrix}\).

= \(2(7)-3(5)=14-15=-1\).

Example 3
Solve by inverse: \(x+2y=5\), \(3x+4y=11\).

Coefficient matrix \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\), \(X=\begin{bmatrix}x\\y\end{bmatrix}\), \(B=\begin{bmatrix}5\\11\end{bmatrix}\).
\(|A|=1(4)-2(3)=-2\).
Adj\((A)=\begin{bmatrix}4&-2\\-3&1\end{bmatrix}\).
\(A^{-1}=\frac{1}{-2}\begin{bmatrix}4&-2\\-3&1\end{bmatrix}\).
\(X=A^{-1}B=\begin{bmatrix}1\\2\end{bmatrix}\).
So \(x=1\), \(y=2\).

46.12 Practice Problems

  1. Find \(A^T\) for \(A=\begin{bmatrix}1&-2&3\\4&0&5\end{bmatrix}\).
  2. If \(A=\begin{bmatrix}2&3\\1&4\end{bmatrix}\) and \(B=\begin{bmatrix}1&0\\5&2\end{bmatrix}\), find \(AB\).
  3. Show that \(AB\neq BA\) for above matrices.
  4. Find determinant of \(\begin{bmatrix}1&2&3\\0&1&4\\5&6&0\end{bmatrix}\).
  5. Solve using inverse: \(2x+y=5\), \(x-y=1\).

46.13 Summary

  • Matrix = array of numbers, with defined size/order.
  • Types include row, column, square, diagonal, identity, symmetric.
  • Operations: addition, multiplication, transpose, inverse.
  • Determinant is key for square matrices.
  • Systems of linear equations can be compactly solved using matrix methods.