Matrices

A Matrix is an organized rectangular array comprising numbers, symbols, or expressions that are systematically arranged in rows and columns. It is typically denoted by uppercase letters and is used in Mathematics and Computer science to represent data or perform operations such as addition, subtraction, multiplication, and more.

1.0Types of Matrices

1. Column Matrix

A matrix is termed a Column Matrix if it contains only one column.

Example: $A=\left[\begin{array}{c}0\\ 3\\ -2\\ \frac{1}{2}\end{array}\right]$

2. Row Matrix

A matrix is considered a Row Matrix if it consists of only one row.

Example - A = [1 5 7]

3. Square Matrix

A square matrix is one where the number of rows equals the number of columns.

Example : $A=\left[\begin{array}{ll}5& 7\\ 1& 3\end{array}\right]$

4. Diagonal Matrix

A square matrix B = [b_ij] of order m × m is classified as a Diagonal Matrix if all its non-diagonal elements are zero. In other words, a matrix B = [b_ij] of size m × m is designated as a diagonal matrix if b_ij = 0 whenever i ≠ j.

$\left[\begin{array}{llll}1& 0& 0& 0\\ 0& 5& 0& 0\\ 0& 0& 7& 0\\ 0& 0& 0& 3\end{array}\right]$

5. Scalar Matrix

A Scalar Matrix is a square matrix where all diagonal elements are equal (scalar), and all non-diagonal elements are zero.

6. Identity Matrix

A square matrix in which ones are on the main diagonal and zeros anywhere else is known as an identity matrix. It is denoted as I_n for an n × n matrix.

$\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

7. Zero Matrix

A zero matrix, denoted as 0, is a matrix in which all elements are zero.

$\left[\begin{array}{lll}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

8. Orthogonal Matrix

An orthogonal matrix is a square matrix where the transpose is equal to its inverse. 

A^T = A^–1 

$\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$

2.0Operation on Matrices

1. Addition of two Matrices

To add two matrices, you add corresponding elements in the same position in each matrix. Here is the general formula for adding two matrices:

If two matrices A = [a_ij] and B=[b_ij] have same order m×n, then their sum A + B is given by:

A + B=[a_ij] + [b_ij] = [a_ij +b_ij]

Example:

$A=\left[\begin{array}{lll}4& 5& 3\\ 7& 8& 9\end{array}\right]B=\left[\begin{array}{lll}7& 6& 2\\ 1& 2& 3\end{array}\right]$

$A+B=\left[\begin{array}{lll}4+7& 5+6& 3+2\\ 7+1& 8+2& 9+3\end{array}\right]=\left[\begin{array}{ccc}11& 11& 5\\ 8& 10& 12\end{array}\right]$

• Addition of matrices follows the following Properties
• Commutative Law: A + B = B + A
• Associative Law: (A + B) + C = A + (B + C)
• Additive Identity: O (Zero Matrix) is the Additive Identity for matrix addition.
• Additive Inverse: –A is the additive inverse of A or negative of A.
  

2.  Multiplication of a matrix by a scalar

Multiplying a matrix by a scalar involves multiplying each element of the matrix by the scalar value. For example, if we have a matrix A and a scalar k, the product kA is obtained by multiplying each element of A by k.

Example:

$A=\left[\begin{array}{lll}4& 5& 3\\ 7& 8& 9\end{array}\right]$

$\mathrm{kA}=\mathrm{k}\times \left[\begin{array}{ccc}4& 5& 3\\ 7& 8& 9\end{array}\right]=\left[\begin{array}{ccc}4\mathrm{k}& 5\mathrm{k}& 3\mathrm{k}\\ 7\mathrm{k}& 8\mathrm{k}& 9\mathrm{k}\end{array}\right]$

• Scalar Multiplication of a matrix follows the following Properties
• k(A +B) = k A + kB,

3. Multiplication of Matrices

Matrix multiplication is a binary operation that combines two matrices to produce a new matrix. For matrix multiplication to be defined the number of columns in matrix A should match the number of rows in matrix B.

Let's consider two matrices, A and B:

$A=\left[\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right]$

$B=\left[\begin{array}{lll}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\end{array}\right]$

To perform matrix multiplication between matrices A and B, we compute each element of the resulting matrix C by taking the scalar product of the corresponding row of matrix A and the respective column of matrix B:

To multiply A and B, we follow the rule that the (i,j)^th element of the resulting matrix C is obtained by taking the scalar product of the i^th row of A and the j^th column of B:

$C=A\cdot B=\left[\begin{array}{lll}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\\ c_{31}& c_{32}& c_{33}\end{array}\right]$

where

c₁₁ = a₁₁⋅b₁₁ + a₁₂⋅b₂₁,

c₁₂ = a₁₁⋅b₁₂ + a₁₂⋅b₂₂,

c₁₃ = a₁₁⋅b₁₃ + a₁₂⋅b₂₃,

c₂₁ = a₂₁⋅b₁₁ + a₂₂⋅b₂₁,

c₂₂ = a₂₁⋅b₁₂ + a₂₂⋅b₂₂,

c₂₃ = a₂₁⋅b₁₃ + a₂₂⋅b₂₃,

c₃₁ = a₃₁⋅b₁₁ + a₃₂⋅b₂₁,

c₃₂ = a₃₁⋅b₁₂ + a₃₂⋅b₂₂,

c₃₃ = a₃₁⋅b₁₃ + a₃₂⋅b₂₃.

• Multiplication of Matrices follows following Properties
• Associative Law: (AB)C = A(BC)

3.0Transpose of Matrices

The transpose of a matrix is generated by interchanging its rows and columns matrix.. For a matrix A with dimensions m×n, the transpose A^T has dimensions n×m.

Let's consider a matrix A with elements a_ij:

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]$

The transpose AT is formed by interchanging rows and columns:

$A^T=\left[\begin{array}{cccc}{a}_{11}& a_{21}& \dots & a_{1m}\\ a_{12}& a_{22}& \dots & a_{m2}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \dots & a_{mn}\end{array}\right]$

In other words, the (i, j)^th element of A^T is the (j, i)^th element of A, i.e.,

( A^T)_ij= A_ji.

The transpose operation is important in various matrix operations and has several properties, such as:

1. (A^T)^T = A: Transposing a matrix twice returns the original matrix.
2. (kA)^T = k(A^T ): Transposing a scalar multiple of a matrix is the same as multiplying the transpose of the matrix by the scalar. 
3. (A+B) ^T =A^T + B^T: Transposing the sum of two matrices is the same as taking the sum of their transposes.
4. (AB)^T = B^TA^T : Transposing the product of two matrices is equivalent to multiplying their transposes in the reverse order.

4.0Symmetric and Skew Symmetric Matrices

1. Symmetric Matrix

A square matrix A = [a_ij] is said to be, symmetric if, a_ij =a_ij ∀ i and j(conjugate elements are equal). 

Hence for symmetric matrix A = A ^T. 

2. Skew Symmetric Matrix

Square Matrix A = [a_ij] is said to be skew symmetric if, a_ij = –a_ij ∀ i and j(the pair of conjugate elements are additive inverse of each other). For a skew symmetric matrix, A = – A^T. 

5.0Invertible Matrices 

A square matrix is A = [aij] is said to be invertible if |A| ≠ 0, and AA^–1 = A^–1A = I.

6.0Important Formula of Matrices

• A(adj.A) = | A | I_n = (adj A) A
•  | adj A | = | A |^n–1 (Thus A (adj A) is always a scalar matrix)
•  adj (adj.A) = | A | ^n–2 A, where |A| ≠ 0
•  |adj (adj.A)| = A, where |A| ≠ 0
• adj (AB) = (adj B) (adj A)
• adj (A^m) = (adj A) ^m,
• adj (KA) = K^n–1 (adj A), K is scalar
• adj (I_n) = I_n, 
•  adj (0) = 0
• A is symmetric ⇒ adj A is also symmetric
• A is diagonal ⇒ adj A is also diagonal
• A is triangular ⇒ adj A is also triangular
• A is singular ⇒ | adj A | = 0

7.0Solved Problems on Matrices

Example 1: If $A=\left[\begin{array}{ll}1& 3\\ 3& 2\\ 2& 5\end{array}\right]$ & $B=\left[\begin{array}{cc}-1& -2\\ 0& 5\\ 3& 1\end{array}\right]$ and A+ B –D = 0 (Zero matrix), then D matrix will be ?

(A) $\left[\begin{array}{ll}0& 2\\ 3& 7\\ 6& 5\end{array}\right]$

(B)$\left[\begin{array}{ll}0& 2\\ 3& 7\\ 5& 6\end{array}\right]$

(C) $\left[\begin{array}{ll}0& 1\\ 3& 7\\ 5& 6\end{array}\right]$

(D) $\left[\begin{array}{cc}0& -2\\ -3& -7\\ -5& -6\end{array}\right]$

Ans. (C)

Solution:

$LetD=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\\ \mathrm{e}& \mathrm{f}\end{array}\right]$

∴ $\text{ A+B-D }=\left[\begin{array}{ll}1& 3\\ 3& 2\\ 2& 5\end{array}\right]+\left[\begin{array}{cc}-1& -2\\ 0& 5\\ 3& 1\end{array}\right]-\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\\ \mathrm{e}& \mathrm{f}\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}1-1-\mathrm{a}& 3-2-\mathrm{b}\\ 3+0-\mathrm{c}& 2+5-\mathrm{d}\\ 2+3-\mathrm{e}& 5+1-\mathrm{f}\end{array}\right]=\left[\begin{array}{ll}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

⇒ –a = 0 ⇒ a = 0, 1 –b = 0 ⇒  b = 1,

⇒  3–c = 0 ⇒ c = 3,7 – d = 0 ⇒ d = 7,

⇒ 5 – e = 0 ⇒  e = 5,6 – f = 0 ⇒ f = 6 

$\therefore \mathrm{D}=\left[\begin{array}{ll}0& 1\\ 3& 7\\ 5& 6\end{array}\right]$

Example 2: If A,B are two matrices such that A+B=$\left[\begin{array}{ll}1& 2\\ 2& 4\end{array}\right]$, A-B=$\left[\begin{array}{cc}3& 2\\ -2& 0\end{array}\right]$, then find AB.

Solution:

Given A+B=$\left[\begin{array}{ll}1& 2\\ 2& 4\end{array}\right]$ …(i)  &

$\Rightarrow \mathrm{A}-\mathrm{B}=\left[\begin{array}{cc}3& 2\\ -2& 0\end{array}\right]$ …(ii)  &

Adding (i) & (ii)

$\Rightarrow 2\mathrm{A}=\left[\begin{array}{ll}4& 4\\ 0& 4\end{array}\right]\Rightarrow \mathrm{A}=\left[\begin{array}{ll}2& 2\\ 0& 2\end{array}\right]$

Subtracting (ii) from (i)

$\Rightarrow 2\mathrm{B}=\left[\begin{array}{cc}-2& 0\\ 4& 4\end{array}\right]\Rightarrow \mathrm{B}=\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]$

Now A B=

$\left[\begin{array}{ll}2& 2\\ 0& 2\end{array}\right]\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]$

$=\left[\begin{array}{ll}2& 4\\ 4& 4\end{array}\right]$

Example 3: If A=$\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]$, show that ${\mathrm{A}}^{\mathrm{k}}=\left[\begin{array}{cc}1+2\mathrm{k}& -4\mathrm{k}\\ \mathrm{k}& 1-2\mathrm{k}\end{array}\right]$ , where k is any positive integer.

Solution: We have 

$\Rightarrow {\mathrm{A}}^2=\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]=\left[\begin{array}{ll}5& -8\\ 2& -3\end{array}\right]=\left[\begin{array}{cc}1+2\times 2& -4\times 2\\ 2& 1-2\times 2\end{array}\right]$

and

$\Rightarrow {\mathrm{A}}^3=\left[\begin{array}{ll}5& -8\\ 2& -3\end{array}\right]\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]=\left[\begin{array}{cc}7& -12\\ 3& -5\end{array}\right]=\left[\begin{array}{cc}1+2\times 3& -4\times 3\\ 3& 1-2\times 3\end{array}\right]$

Thus, it is true for indices 2 and 3. Now assume

$\Rightarrow A^{\mathrm{k}}=\left[\begin{array}{cc}1+2\mathrm{k}& -4\mathrm{k}\\ \mathrm{k}& 1-2\mathrm{k}\end{array}\right]$

Then,

$\Rightarrow {\mathrm{A}}^{\mathrm{k}+1}=\left[\begin{array}{cc}1+2\mathrm{k}& -4\mathrm{k}\\ \mathrm{k}& 1-2\mathrm{k}\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$

$=\left[\begin{array}{cc}3+2\mathrm{k}& -4(\mathrm{k}+1)\\ \mathrm{k}+1& -1-2\mathrm{k}\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}1+2(\mathrm{k}+1)& -4(\mathrm{k}+1)\\ \mathrm{k}+1& 1-2(\mathrm{k}+1)\end{array}\right]$

Example 4: If A is symmetric as well as skew symmetric matrix, then A is-

(A) diagonal matrix (B) null matrix

(C) triangular matrix (D) none of these

Ans. (B)

Solution: Let A = [a_ij] Since A is skew symmetric a_ij = –a_ji 

for  i = j, a_ii = –a_ii ⇒ a_ii =0

for i ≠  j, a_ij = – a_ji [∵A is skew symmetric ] and a_ij = a_ji [∵ A is symmetric]

∴ a_ij = 0 for all i ≠ j

So, a_ij = 0 for all ‘i’ and ‘j’ i.e. A is a null matrix.

Example 5: If A=$\left[\begin{array}{lll}2& 0& 0\\ 2& 2& 0\\ 2& 2& 2\end{array}\right]$, then adj (adj A) is equal to

(A) $8\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$

(B) $16\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$

(C) $64\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$

(D) None of these

Ans. (B)

Solution: 

|A|=$\left[\begin{array}{lll}2& 0& 0\\ 2& 2& 0\\ 2& 2& 2\end{array}\right]=8$

Now adj (adj A) = |A|^3–2A

=$8\left[\begin{array}{lll}2& 0& 0\\ 2& 2& 0\\ 2& 2& 2\end{array}\right]=16\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$

8.0Solved Questions on Matrices

Q1. What is an invertible matrix?

Ans: An invertible matrix, also known as a nonsingular matrix, is a square matrix that has an inverse. An invertible matrix A satisfies the condition that there exists another matrix A⁻¹ such that AA⁻¹ =A⁻¹A = I, where I represent the identity matrix.

Q2. How do you find the determinant of a matrix?

Ans: The determinant of a square matrix can be found using various methods such as cofactor expansion.

For a 2 × 2 matrix $\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$ , the determinant is given by ad−bc. For larger matrices, the process involves recursive calculations based on cofactors.