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Matrix: Types

Transpose of a Matrix:

 

Matrix formed by exchanging rows and corresponding columns of the original matrix. Transpose of a matrix A:

Transpose
GATE matrix algebra
GATE matrix algebra

Hence; (a)ij of [A] = (a)ji of transpose of [A], where (a)ij is element in ith row and jth column and (a)ji is element in jth row and ith column.

* Value of determinant of [A] and transpose of [A] is same i.e. lAl = lTrans. Al

Null Matrix:

 

Matrix having all the elements as zero.

Null Mat
GATE matrix algebra

This matrix transforms all vectors in the space to zero vector i.e. at the origin.

Square Matrix:

Matrix having equal number of rows and columns. A 2 x 2 matrix is:

Sqr Mat
GATE matrix algebra

2 x 2 square matrix transforms a 2-D vector such that it remains in 2-D plane only. Likewise, a 3 x 3 square matrix transforms a 3-D vector such that remains in 3-D space only. Whereas a non-square matrix may transform a 2-D vector into 3-D vector and vice-versa.

Diagonal Matrix:

A square matrix having only non-zero diagonal elements. All other elements of the matrix are zero.

diag mat
GATE matrix algebra

A diagonal matrix is called 'Scalar Matrix' when a = b. A scalar matrix only stretches or shrinks a vector along its span.

A diagonal matrix is called a 'Unit/Identity Matrix' if a = b = 1. Unit matrix is represented by I and does nothing to the vector after transformation. A 3x3 unit matrix is:

scala mat
uni mat
GATE matrix algebra

​​Symmetric and Skew-Symmetric Matrix:

A square matrix is called 'Symmetric' if

(a)ij  = (a)ji

Hence; Trans.[A] = [A]

A 3 x 3 symmetric matrix is:

sym skew mat
GATE matrix algebra

For a 'Skew-Symmetric Matrix'

(a)ij  = -(a)ji and (a)ii = 0

Hence;Trans.[A] = -[A]

A 3 x 3 skew-symmetric matrix is:

GATE matrix algebra

Orthogonal Matrix:

A square matrix satisfying the condition

ortho mat
GATE matrix algebra

An orthogonal matrix transforms a space such that axes always remains perpendicular to each other and magnitude of unit vectors remains unchanged (rotation/reflection). Consider an orthogonal matrix:

GATE matrix algebra

For the above matrix

GATE matrix algebra

so i(transformed) and j(transformed) are perpendicular to each other and their magnitude is equal to 1.

* Value of determinant of an orthogonal matrix is equal to 1 or -1 i.e. lAl = 1 or -1.

Idempotent Matrix:

 

For an idempotent matrix

idem mat
GATE matrix algebra

Nilpotent Matrix:

 

​For a nilpotent matrix

nilpo mat
GATE matrix algebra

k is a positive integer.

Least positive integer for which above value is true is called index of the matrix.

Involuntary Matrix:

For an involuntary matrix

invol mat
GATE matrix algebra

It means matrix is inverse of itself.

​​Singular Matrix:

A matrix is called a singular matrix if its determinant is zero i.e. lAl = 0.

​Triangular Matrix:

A square matrix having all elements below the main diagonal as zero is called 'Upper Triangular (UT) Matrix'. For upper triangular matrix (a)ij = 0 for all i>j.

sing mat
traing mat
GATE matrix algebra

A square matrix having all elements above the main diagonal as zero is called 'Lower Triangular (LT) Matrix'. For lower triangular matrix (a)ij = 0 for all i<j.

GATE matrix algebra

* Transpose of a LT Matrix is a UT matrix.

* Determinant of triangular matrix is equal to product of diagonal elements.

Complex Matrices:

Matrix whose elements may be complex numbers.

a) Conjugate of a Matrix:

Matrix formed by conjugates of complex elements of original matrix.

Comp mat
GATE matrix algebra

b) Conjugate Transpose:

Transpose of conjugate of a complex matrix.

c) Unitary Matrix:

A complex square matrix is said to be Unitary when product of the matrix (A) with its conjugate transpose (A*) is equal to an identity matrix i.e. AA* = I

d) Hermitian Matrix:

A complex square matrix is called hermitian if (a)ij  = conjugate of (a)ji. A 2 x 2 Hermitian matrix is:

GATE matrix algebra

For a Hermitian matrix A = A*. Diagonal elements of the matrix will always be either real or zero.

 

e) Skew-Hermitian Matrix

A complex square matrix is called Skew-Hermitian if (a)ij = - conjugate of (a)ji. A 2 x 2 Skew-Hermitian matrix is:

GATE matrix algebra

For a Skew-Hermitian matrix A = -A*. Diagonal elements of the matrix will always be either zero or pure imaginary.

Example:

GATE 2016: A real square matrix A is called skew-symmetric if

Gate ques
GATE 2016 solved
GATE 2016 solved
GATE 2016 solved
GATE 2016 solved

Answer: C

Example:

GATE 2009: For a Matrix [M] as given below, the transpose of the matrix is equal to the inverse of the matrix. The value of x is:

GATE 2009 solved

A) -4/5

B) -3/5

C) 3/5

D) 4/5

Solution:

Since Trans. of [M]   = Inv. of [M], so the matrix is an orthogonal matrix. Hence column vectors will be perpendicular to each other and their magnitude will be equal to 1.

Column 1 vector C1 = 3/5 i + x j

Column 2 vector C2 = 4/5 i + 3/5 j

For perpendicularity of C1 and C2: (3/5)*(4/5) + (x)*(3/5) = 0

Hence x = -4/5

The correct answer is Option 'A'.

Example:

GATE 2006: Multiplication of matrices E & F is G. Matrix E and G are as given below. What is the matrix F?

GATE 2006 solved
GATE 2006 solved
GATE 2006 solved
GATE 2006 solved
GATE 2006 solved
GATE 2006 solved

Solution:

For matrix E: Column vector C1, C2 and C3 are perpendicular to each other (vector dot product is zero) and magnitude of each column vector is 1. Hence matrix E is Orthogonal matrix.

For orthogonal Matrix E x (Trans. of E) = I

Here it is given as E x F = G, where G is identity matrix.

Hence F must be transpose of E.

The correct answer is option 'C'.

Example:

GATE 2004: For which value of x, the matrix given below will become singular:

GATE 2004 solved

a) 4

b) 6

c) 8

d) 12

 

Solution:

For a singular matrix lAl = 0

8*(0-12) - x*(0-24) + 0 = 0

-96 + 24*x = 0

x = 4

The correct answer is option 'A'.

Example:

GATE 2017: Consider the matrix P as given below. Which one of the following statements about P is incorrect.

A) Determinant of P is equal to 1.

B) P is orthogonal.

C) Inverse of P is equal to its transpose.

D) All eigenvalue of P are real numbers.

GATE 2017 engineering mathematics

Solution:

For matrix P Column vector C1, C2 and C3 are perpendicular to each other (vector dot product is zero) and magnitude of each column vector is 1. Hence matrix P is Orthogonal matrix.

For an orthogonal matrix:

|P| = 1 and 'Inverse of P = Transpose of P'

Hence option D is the correct answer.

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