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 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,

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:

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

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 symmetric if

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

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 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.

A square matrix A of order m × m is said to be invertible (or non-singular) if there exists another square matrix B of the same order such that

where I is the identity matrix of order m. Then B is called the inverse matrix of A, and it is denoted by \[A^{-1}\].

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.

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.

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. 

The inverse of a square matrix, if it exists, is unique.

Proof: Let A = [aᵢⱼ] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C.

Since B is the inverse of A

AB = BA = I ...(1)

Since C is also the inverse of A

AC = CA = I ...(2)

Thus

B = BI = B(AC) = (BA)C = IC = C

So B = C, which means the inverse of A is unique.

If A and B are invertible matrices of the same order, then
(AB)⁻¹ = B⁻¹A⁻¹.

Proof: From the definition of the inverse of a matrix, we have

(AB)(AB)⁻¹ = I

or A⁻¹(AB)(AB)⁻¹ = A⁻¹ (Pre multiplying both sides by A⁻¹)

or (A⁻¹A)B(AB)⁻¹ = A⁻¹ (Since A⁻¹A = I)

or IB(AB)⁻¹ = A⁻¹

or B(AB)⁻¹ = A⁻¹

or B⁻¹B(AB)⁻¹ = B⁻¹A⁻¹

or I(AB)⁻¹ = B⁻¹A⁻¹

Hence (AB)⁻¹ = B⁻¹A⁻¹

• 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.

• 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.

• 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.

• 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.

• Invertible matrices must be square.

• The inverse satisfies \[AA^{-1} = A^{-1}A = I\].

• The inverse, if it exists, is unique.

• For invertible matrices A and B of the same order: \[(AB)^{-1} = B^{-1}A^{-1}\].

• Rectangular matrices do not have inverses.

• If B is the inverse of A, then A is also the inverse of B.