Revision: Mathematics >> Matrices CUET (UG) Matrices

A matrix is a rectangular arrangement of numbers arranged in rows and columns, enclosed in brackets [ ] or parentheses ( ).

Elements (Entries) of a Matrix

• Each number in a matrix is called an element (or entry).

Rows and Columns

• Horizontal lines → rows
• Vertical lines → columns

Order of a Matrix

• Order = number of rows × number of columns
• Written as m × n and read as “m by n”

Two matrices are equal if and only if:

1. They have the same order (same number of rows and columns), and
2. Their corresponding elements are equal.

Example:

\[A=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\mathrm{and} B=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\]

Let \[A = [a_{ij}]\] and \[B = [b_{ij}]\] be two matrices of the same order \[m \times n\].

Their sum \[C = A + B\] is defined as the matrix \[[c_{ij}]\] of order \[m \times n\], where

Let \[A = [a_{ij}]\] and \[B = [b_{ij}]\] be matrices of the same order \[m \times n\].

Their difference \[D = A - B\] is defined as the matrix \[[d_{ij}]\] where

Equivalently,

The transpose of a matrix is obtained by interchanging its rows and columns.

• If a matrix is A, its transpose is denoted by A^T

• If A is of order m × n, then
  A^T is of order n × m

• First row of A becomes first column of A^T, and so on.

A square matrix A = [aᵢⱼ]ₙ×ₙ is symmetric if Aᵀ = A
i.e., aᵢⱼ = aⱼᵢ for all i and j.

Skew-Symmetric Matrix: A square matrix A = [aij] n×n is skew-symmetric if Aᵀ = −A i.e., aij = −aji for all i and j.

Skew-Symmetric Matrix: A square matrix A = [aij] n×n is skew-symmetric if Aᵀ = −A i.e., aij = −aji for all i and j.

A square matrix A = [aᵢⱼ]ₙ×ₙ is symmetric if Aᵀ = A
i.e., aᵢⱼ = aⱼᵢ for all i and j.

If A and B are symmetric matrices.

∴ A’ = A and B’ = B

(AB – BA) = (AB)’ – (BA)’   ...[∵ (X – Y) = X’ – Y’]

= B’A’ – A’B’   ...[∵ (XY) = Y’X’]

= BA – AB   ...[∵ B’ = B, A’ = A]

= –(AB – BA)

∴ AB – BA is a skew symmetric matrix.

If A and B are symmetric matrices.

∴ A’ = A and B’ = B

(AB – BA) = (AB)’ – (BA)’   ...[∵ (X – Y) = X’ – Y’]

= B’A’ – A’B’   ...[∵ (XY) = Y’X’]

= BA – AB   ...[∵ B’ = B, A’ = A]

= –(AB – BA)

∴ AB – BA is a skew symmetric matrix.

• Matrix: A rectangular array of elements.

• Element: An entry inside a matrix.

• Order: Size of a matrix written as rows × columns.

• Row: Horizontal set of elements.

• Column: Vertical set of elements.

• a_ij​: Element in the i-th row and j-th column.

• Equality of matrices is possible only when the order is the same.

• Corresponding elements must be compared position by position.

• If even one corresponding entry differs, the matrices are not equal.

• Matrices must be of same order for addition and subtraction.

• \[A + B = [a_{ij} + b_{ij}]\].

• A - B = A + (-B).

• Addition is commutative: A + B = B + A.

• Addition is associative: (A + B) + C = A + (B + C).

• Zero matrix is additive identity: A + O = A.

• Negative of a matrix is additive inverse: \[A + (-A) = O\].

• If order differs \[\rightarrow\] operation not defined.