Revision: Applied Mathematics >> Algebra CUET (UG) Algebra

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 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 \[A = [a_{ij}]_{n \times n}\] is called skew-symmetric if \[A^T = -A\]

i.e.,\[a_{ij} = -a_{ji}\] for all i and j.

A square matrix \[A = [a_{ij}]_{n \times n}\] is called symmetric if

i.e., \[a_{ij} = a_{ji}\] 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.

Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Proof: Let A be a square matrix, then we can write

\[\mathrm{A=\frac{1}{2}(A+A^{\prime})+\frac{1}{2}(A-A^{\prime})}\]

From Theorem 1, we know that (A + A′) is a symmetric matrix and (A − A′) is a skew-symmetric matrix.

Multiplying by \[\frac{1}{2}\] does not change these properties.

Since for any matrix A, (kA)′ = kA′, it follows that \[\frac{1}{2}(\mathrm{A}+\mathrm{A}^{\prime})\] is symmetric matrix and \[\frac{1}{2}(\mathrm{A}-\mathrm{A}^{\prime})\] is skew symmetric matrix.

Thus, any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. 

Theorem 1: For any square matrix A with real number entries, A + A′ is a symmetric matrix and A − A′ is a skew-symmetric matrix.

Proof:

Part 1: Symmetric Matrix
Let B = A + A′, then

Take transpose on both sides:
B′ = (A + A′)′
= A′ + (A′)′ (as (A + B)′ = A′ + B′)
= A′ + A (as (A′)′ = A)
= A + A′ (as A + B = B + A)
= B

Therefore, B = A + A′ is a symmetric matrix

Part 2: Skew-Symmetric Matrix

Now let
C = A − A′

C′ = (A − A′)′ = A′ − (A′)′ (Why?)
= A′ − A (Why?)
= −(A − A′) = −C

Therefore
C = A − A′ 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.

• Transpose = interchange rows and columns.

• If A is \[m \times n\], then A' is \[n \times m\].

• Standard notation: A' or \[A^T\].

• Key properties: (A')' = A, (kA)' = kA', (A + B)' = A' + B', (AB)' = B'A'.

• A square matrix is symmetric if \[A^T = A\].

• A square matrix is skew-symmetric if \[A^T = -A\].

• In a skew-symmetric matrix, all diagonal elements are zero.

• For any square matrix A:

  • \[A + A^T\] is symmetric.

  • \[A - A^T\] is skew-symmetric.

• Any square matrix A can be written as

• The decomposition into symmetric and skew-symmetric parts is unique.