Figure Matrix Tips and Tricks for Government Exams

Table of Contents
1. What is a matrix ?
2. Transpose of a matrix
3. Addition of matrices
4. Subtraction of matrices
5. Multiplication of matrices
6. Determinant of a matrix
7. Inverse of a matrix
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What is a matrix ?

A matrix is a rectangular array of numbers, symbols or expressions arranged in rows and columns. Each item in a matrix is called an entry or an element. The size (or order) of a matrix is given as P × Q, where P is the number of rows and Q is the number of columns. For example, a matrix of order 2 × 3 has two rows and three columns.

• Row vector - a matrix with a single row. Example: 1 × n = [4 6 9].
• Column vector - a matrix with a single column. Example: n × 1 = see image below.

• Square matrix - a matrix with the same number of rows and columns; order n × n. Example: see image below.
  

Transpose of a matrix

The transpose of a matrix A is obtained by interchanging its rows and columns. The transpose of A is denoted by A^T. If A = [a_ij], then the transpose A^T = [a_ji], i.e., the element at row i, column j in A becomes the element at row j, column i in A^T.

Addition of matrices

Two matrices can be added (or subtracted) only if they have the same order (same number of rows and same number of columns). If A = [a_ij] and B = [b_ij] are both m × n matrices, then their sum A + B is an m × n matrix whose entries are given by (A + B)_ij = a_ij + b_ij.

Subtraction of matrices

Subtraction follows the same rule as addition: two matrices must have the same order. If A and B are both m × n, then A - B is the m × n matrix with entries (A - B)_ij = a_ij - b_ij.

Multiplication of matrices

There are two commonly used types of multiplication involving matrices: scalar multiplication and matrix multiplication.

• Scalar multiplication - multiplying every entry of a matrix by a single number (scalar). If k is a scalar and A is a matrix, then kA is formed by multiplying each entry of A by k.
• Matrix multiplication - to multiply two matrices, the number of columns of the first matrix must equal the number of rows of the second matrix. If A has order m × p and B has order p × n, then the product AB exists and has order m × n. The (i, j) entry of AB is the sum of products of corresponding entries from the i-th row of A and the j-th column of B.
• Non-commutativity - in general, matrix multiplication is not commutative: for matrices of compatible orders, AB need not equal BA. Thus CD ≠ DC in general.

Example (dimension rule): if the first matrix is 3 × 2 and the second is 2 × 3, then the product exists and the resulting matrix has order 3 × 3, because the inner dimensions (2 and 2) match and the outer dimensions (3 and 3) give the result size.

Determinant of a matrix

The determinant is a scalar value that can be computed for a square matrix and provides important information about the matrix, such as whether it is invertible. Only square matrices have determinants.

For a 2 × 2 matrix A = [ [a b], [c d] ], the determinant is det(A) = ad - bc.

For a 3 × 3 matrix, there are several methods to compute the determinant, including expansion by minors/cofactors and Sarrus' rule (a mnemonic useful for 3 × 3 matrices). See the illustrations below for the formulae and worked pattern.

Inverse of a matrix

The inverse of a square matrix A is a matrix A^-1 such that AA^-1 = A^-1A = I, where I is the identity matrix of the same order. A matrix has an inverse only if it is square and its determinant is non-zero (i.e., det(A) ≠ 0).

For a 2 × 2 matrix A = [ [a b], [c d] ] with det(A) = ad - bc ≠ 0, the inverse is

A^-1 = (1 / det(A)) · [ [d -b], [-c a] ]

In general, the inverse of an n × n matrix can be computed using the adjoint (adjugate) and determinant:

A^-1 = (1 / det(A)) · adj(A)

See the images below for worked examples and formulae for finding inverses.