Introduction to Matrices | Engineering Mathematics - Engineering Mathematics PDF Download

What is a Matrix?

A matrix represents a collection of numbers arranged in an order of rows and columns. It is necessary to enclose the elements of a matrix in parentheses or brackets.

A matrix with 9 elements is shown below.
This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number. For example, a₂₃ = 6 

Order of a Matrix 

The order of a matrix is defined in terms of its number of rows and columns.

Order of a matrix = No. of rows × No. of columns

Therefore Matrix [M] is a matrix of order 3 × 3.

Transpose of a Matrix

The transpose [M]^T of an m x n matrix [M] is the n x m matrix obtained by interchanging the rows and columns of [M].

if A = [a_ij] mxn , then AT = [b_ij] nxm where b_ij = a_ji 

Properties of transpose of a matrix

• (AT)^T = A
• (A + B)^T = A^T + B^T
• (AB)^T = B^TA^T

Singular and Nonsingular Matrix 

1. Singular Matrix: A square matrix is said to be a singular matrix if its determinant is zero i.e. |A| = 0
2. Nonsingular Matrix: A square matrix is said to be a non-singular matrix if its determinant is non-zero.

Matrix Addition and Multiplication: Properties

• Commutative Property of Addition: For any two matrices A and B of the same order, their sum is independent of the order in which they are added. Mathematically, A + B = B + A.
• Associative Property of Addition: When three matrices A, B, and C are added, the way in which they are grouped does not affect the sum. This means that (A + B) + C = A + (B + C).
• Non-Commutative Property of Multiplication: For most matrices, the product of A and B is not the same as the product of B and A. In other words, AB ≠ BA in general.
• Associative Property of Multiplication: When multiplying three matrices A, B, and C, the way in which they are grouped does not affect the product. This means that (AB)C = A(BC).
• Distributive Property: Matrix multiplication distributes over addition. This means that if A is multiplied by the sum of B and C, it is the same as multiplying A by B and then by C separately. Mathematically, A(B + C) = AB + AC.

Types of Matrices

1. Square Matrix:. matrix is called a square matrix if it has the same number of rows and columns. For example, a 3x3 matrix is square because it has 3 rows and 3 columns.

2. Symmetric Matrix:. square matrix is symmetric if its transpose is equal to the original matrix. In other words, if A is the matrix, then A^T = A. This means that the elements are mirrored along the diagonal.

3. Skew-Symmetric Matrix:. skew-symmetric matrix is a square matrix whose transpose is equal to its negative. This can be represented as A^T = -A. This means that the elements are also mirrored along the diagonal, but with opposite signs.

4. Diagonal Matrix:. diagonal matrix is a square matrix where all the non-diagonal entries are zero. This means that only the elements on the diagonal can be non-zero. For example, in a 3x3 diagonal matrix, A₁₁, A₂₂, and A₃₃ can be non-zero, but A₁₂, A₁₃, A₂₁, A₂₃, A₃₁, and A₃₂ must be zero.

5. Identity Matrix: An identity matrix is a special type of square matrix where all the diagonal elements are ones and all other elements are zeros. It is denoted as I. For example, in a 3x3 identity matrix, the elements I₁₁, I₂₂, and I₃₃ are 1, and all other elements are 0.

6. Orthogonal Matrix:. matrix is orthogonal if the product of the matrix and its transpose is equal to the identity matrix. This can be represented as AA^T = A^TA = I. This means that the rows and columns of the matrix are orthogonal unit vectors.

7. Idempotent Matrix:. matrix is idempotent if when it is multiplied by itself, it gives the same matrix. This can be represented as A² = A. For example, if A is a 2x2 matrix, and A² = A, then A is idempotent.

8. Involutory Matrix:. matrix is involutary if when it is multiplied by itself, it gives the identity matrix. This can be represented as A² = I. This means that the matrix is its own inverse.

Note: It is important to note that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix. This can be represented as A = 1/2 (A^T + A) + 1/2 (A – A^T). This means that any square matrix can be decomposed into two parts: one that is symmetric and one that is skew-symmetric.

Adjoint of a Square Matrix

The adjoint of a matrix A is the transpose of the cofactor matrix of A

Properties of Adjoint 

1. A(Adj A) = (Adj A) A = |A| I_n
2. Adj(AB) = (Adj B). (Adj A)
3. |Adj A| = |A|^{n - 1}
4. Adj(kA) = k^{n - 1} Adj(A)
5. |adj(adj(A))| = |A|^(n - 1)^2
6. adj(adj(A)) = |A|^(n - 2)     *  A
7. If A = [L, M, N] then adj(A) = [MN, LN, LM]
8. adj(I) = I

Where, “n = number of rows = number of columns”

Inverse of a Square Matrix 

The inverse of a Square Matrix

Here |A| should not be equal to zero, which means matrix A should be non-singular. 

Properties of Inverse

1. (A-¹)^-1 = A 
2. (AB)^-1 = B^-1A^-1 
3. Only a non-singular square matrix can have an inverse. 

Where should we use the inverse matrix? 

If you have a set of simultaneous equations:

7x + 2y + z = 21

3y – z = 5

-3x + 4y – 2x = -1

As we know when AX = B, then X = A^-1B so we calculate the inverse of A and by multiplying it B, we can get the values of x, y, and z.

Trace of a matrix

The Trace of a matrix is denoted as tr(A) which is used only for square matrix and equals the sum of the diagonal elements of the matrix. Remember trace of a matrix is also equal to the sum of eigen value of the matrix.

For example:

Solved Examples

Q1: For matrices of the same dimension M, N, and scalar c, which one of these properties DOES NOT ALWAYS hold?
(a) (M^T)^T = M
(b) (cM)^T = c(M)^T 
(c) (M + N)^T = M^T + N^T 
(d) MN = NM
Ans: (d)
Sol: Let us consider two 2 × 2 Matrices (same dimension) as shown:

Similarly, N × M gives:

We observe that (M × N)_2×2  ≠ (N × M)_2×2, even if the dimensions of the two matrices are equal.
But if we take two 2 × 2 Identity Matrices (same dimension), the product will be commutative, i.e. if:

(M × N)2×2  = (N × M)2×2
We, therefore, conclude that (M × N)2×2  IS NOT ALWAYS EQUAL TO (N × M)2×2

Q2: For A = the determinant of A^TA^-1 is: 
(a) sec² x
(b) cos 4x
(c) 1
(d) 0 
Ans: (c)
Sol: 

Q3: Let X be a square matrix. Consider the following two statements on X.
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?
(a) I implies II; II does not imply I.
(b) II implies I; I does not imply II.
(c) I does not imply II; II does not imply I.
(d) I and II are equivalent statements.
Ans: (d)
Sol: I implies II means ≡ I → II

If X^-1 then |X| ≠ 0 also |Adj X| = |X|^{n - 1} then |Adj X| ≠ 0
If X^-1 then |X| ≠ 0
I implies II and II implies I
∴ Both I and II are equivalent 

Q4: The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?
(a) PQ = I but QP ≠ I
(b) QP = I but PQ ≠ I
(c) PQ = I and QP = I
(d) PQ – QP = I
Ans: (c)
Sol: Given, P = Q^-1
Post multiply by Q
PQ = Q^-1Q (we know Q^-1Q = I)
PQ = I
Similary, pre-multiply by Q
QP = QQ^-1
QP = I (QQ^-1 = I)
So, PQ = I and QP = I