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:

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

The negative of a matrix A, denoted by -A, is defined as the scalar multiple \[-1 \cdot A\].

• So, if \[A = [a_{ij}]\], then \[-A = [-a_{ij}]\]

• Adding a matrix to its negative gives the zero matrix: A + (-A) = O

where O is the zero matrix of the same order as A.

Let \[A = [a_{ij}]_{m \times n}\] be a matrix and k be a real number (scalar).

Then the scalar multiple of A by k is the matrix kA defined as:

That is, each entry of A is multiplied by the scalar k.

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,

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 \[A = [a_{ij}]_{m \times n}\] be a matrix and k be a real number (scalar).

Then the scalar multiple of A by k is the matrix kA defined as:

That is, each entry of A is multiplied by the scalar k.

The negative of a matrix A, denoted by -A, is defined as the scalar multiple \[-1 \cdot A\].

• So, if \[A = [a_{ij}]\], then \[-A = [-a_{ij}]\]

• Adding a matrix to its negative gives the zero matrix: A + (-A) = O

where O is the zero matrix of the same order as A.

Let \[A = [a_{ij}]\] be an \[m \times n\] matrix and \[B = [b_{jk}]\] be an \[n \times p\] matrix.

Then the product C = AB is an \[m \times p\] matrix \[C = [c_{ik}]\], where each entry \[c_{ik}\] is given by:

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.

• Scalar multiplication: \[kA = [ka_{ij}]\].

• Negative of a matrix: -A = (-1)A.

• Order of matrix does not change after scalar multiplication.

• k(A + B) = kA + kB.

• (k + l)A = kA + lA.

• k(lA) = (kl)A.

• \[0 \cdot A = O\], \[1 \cdot A = A\].

• 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.

• Scalar multiplication: \[kA = [ka_{ij}]\].

• Negative of a matrix: -A = (-1)A.

• Order of matrix does not change after scalar multiplication.

• k(A + B) = kA + kB.

• (k + l)A = kA + lA.

• k(lA) = (kl)A.

• \[0 \cdot A = O\], \[1 \cdot A = A\].

• Matrix multiplication is row-by-column, not term-wise.

• Product AB exists only if columns of A = rows of B.

• If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].

• In general, \[AB \neq BA\], and sometimes one product may not even be defined.

• Matrix multiplication is associative and distributive over addition.

• Identity matrix acts as a multiplicative identity: AI = IA = A.

• Zero matrix absorbs multiplication: AO = OA = O.