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# Matrices Interview questions and Answers

• February 17, 2023
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• ### What is the characteristic equation of a Matrix:

• a) det(A - λI) = 1
• b) det(A - λI) = 0
• c) A = λI
• d) A = Iλ

Answer - b) det(A - λI) = 0

"Eigen vector of a matrix A is a vector represented by a matrix X. The product of matrix A and vector X, the direction of the output matrix remains same as vector X.
formula: AX = λX
where, A is any arbitrary matrix,
λ are Eigen values and, X is an Eigen vector.

The equation used to find the Eigenvalues of a matrix is called characteristic equation of the matrix. Also known as characteristic polynomial.
For a square matrix A and, λ being any scalar, the characteristic equation of a matrix A is '|A - λI| = 0'

- The equation hints that the matrix (A - λI)X = 0 takes X into the 0 vector. This also means that the (A - λI) will not have an inverse so that its determinant must be 0.

- det(A - λI) = 0 is the characteristic of the matrix

- The trace of a square matrix - Tr(A) = Sum(diagonal elements)"

• ### Cayley-Hamilton theorem in Linear algebra is applicable to:

• a) Orthogonal Matrix
• b) Inverse Matrix
• c) Identity Matrix
• d) Square Matrix

"C.Every square matrix with real or complex numbers satisfies its characteristic (eigen values) equation. A square matrix when transformed into a polynomial equation (det(A - λI)) can be equated to 0.

|A - λI| = 0

example:
A = 1 2
-1 2

A - λI
[1 2 [1 0
-1 2] - λ* 0 1]

|A - λI| will be a polynomial equation with degree equal to n (A is a n*n matrix)
The polynomial equation: λ^2 - 3λ + 4 (degree 2, which is equal to dimension of Matrix A 2*2) equals to zero.
λ^2 - 3λ + 4 = 0"

• ### Diagonalization of matrix:

• a) A a n*n matrix has n linearly independent eigenvectors.
• b) A is similar to a diagonal matrix.
• c) A = PDP^−1 for an invertible matrix P and diagonal matrix D.
• d) All the Above.

Answer - d) All the Above.

"The process of representing a square matrix and converting it into a special type of diagonal matrix that shares the same fundamental properties as the underlying matrix.
Diagonalizing can also be seen as a process of finding the matrix's eigenvalues, which are the diagonal elements of the matrix."

• ### Which of the following is not a square matrix?

• a) 3x3 matrix
• b) 2x3 matrix
• c) 4x3 matrix
• d) 1x2 matrix

"A square matrix has an equal number of rows and columns,.
i.e., the dimension of a square matrix is n x n. Therefore, option D) 1x2 matrix is not a square matrix as it has different number of rows and columns."

• ### Which of the following is a diagonal matrix?

• a) [[1,0,0], [0,2,0], [0,0,3]]
• b) [[1,2,3], [4,5,6], [7,8,9]]
• c) [[1,0], [0,1], [1,0]]
• d) [[0,0], [0,0], [0,0]]

Answer - a) [[1,0,0], [0,2,0], [0,0,3]]

"A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero. Therefore, option a) is a diagonal matrix."

• ### If A is a 3x3 matrix and B is a 3x2 matrix, what will be the dimension of AB?

• a) 3x2
• b) 2x3
• c) 3x3
• d) 2x2

"To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
Here, A has 3 rows and 3 columns, and B has 3 rows and 2 columns. So, the resulting matrix AB will have 3 rows and 2 columns."

• ### Which of the following is not a property of the determinant of a matrix?

• a) It is a scalar value
• b) It can be negative
• c) It is equal to the sum of its diagonal elements
• d) It is equal to the product of its eigenvalues

Answer - c) It is equal to the sum of its diagonal elements

"The determinant of a matrix is a scalar value that can be positive, negative, or zero. It is not equal to the sum of its diagonal elements. However, the sum of the diagonal elements of a matrix is known as the trace of the matrix."

• ### If the determinant of matrix A is zero, what can we say about its inverse?

• a) It exists and is unique
• b) It exists but may not be unique
• c) It does not exist
• d) It is always the zero matrix

Answer - c) It does not exist

"If the determinant of a matrix A is zero, then its inverse does not exist. This is because the inverse of a matrix can be found only if the determinant is not equal to zero."

• ### Which of the following is not a type of matrix?

• a) Identity matrix
• b) Diagonal matrix
• c)Scalar matrix
• d) Octagonal matrix

"Octagonal matrix is not a type of matrix. The other options are types of matrices. "

• ### Which of the following matrices is not a square matrix?

• a) [[1,2], [3,4]]
• b) [[2,3,4], [5,6,7], [8,9,10]]
• c) [[1,2,3], [4,5,6]]
• d) [[-1,0,1], [2,3,-2], [4,1,1]]

"A square matrix has the same number of rows and columns.
The matrices in options a), b), and d) are all square matrices, whereas the matrix in option c) is not a square matrix."

• ### If A is a 3x4 matrix and B is a 4x2 matrix, what is the dimension of the product AB?

• a) 3x2
• b) 3x4
• c) 4x2
• d) 3x3

"The product of two matrices A and B is defined only if the number of columns of A is equal to the number of rows of B. Here, A has 4 columns and B has 4 rows, so their product AB will have 3 rows and 2 columns."

• ### Which of the following is a scalar matrix?

• a) [[1,0], [0,1]]
• b) [[2,0], [0,2]]
• c) [[1,2], [3,4]]
• d) [[0,0], [0,0]]

"A scalar matrix is a diagonal matrix where all the diagonal elements are equal. The matrix in option b) is a scalar matrix with diagonal elements equal to 2."

• ### Which of the following is a symmetric matrix?

• a) [[1,2], [2,1]]
• b) [[1,2], [3,4]]
• c) [[1,0], [0,1]]
• d) [[0,1], [1,0]]

"A matrix is symmetric if it is equal to its transpose. The matrix in option a) is symmetric because it is equal to its transpose."

• ### Which of the following is not a type of matrix multiplication?

• a) Dot product
• b) Scalar multiplication
• c) Cross product
• d) Matrix product

"Cross product is a type of operation that is defined only for vectors and not for matrices. The other options are all types of matrix multiplication."

• ### Which of the following matrices is not invertible?

• a) [[1,2], [3,4]]
• b) [[1,0], [0,1]]
• c) [[2,4], [1,2]]
• d) [[1,1], [2,2]]

"A matrix is invertible if its determinant is not equal to zero.
The determinant of matrix a) is equal to -2, which is not equal to zero, so it is invertible. However, the determinant of matrix
b) is equal to 1, the determinant of matrix
c) is equal to 0, and the determinant of matrix
is equal to 0. Therefore, matrices
, c), and d) are not invertible.>"

• ### Which of the following is a row matrix?

• a) [[1,2,3]]
• b) [[1], [2], [3]]
• c) [[1,2], [3,4]]
• d) [[1]]

"A row matrix is a matrix with only one row. The matrix in option a) is a row matrix."

• ### Which of the following matrices is skew-symmetric?

• a) [[1,2], [2,1]]
• b) [[0,1], [-1,0]]
• c) [[1,0,0], [0,1,0], [0,0,1]]
• d) [[-1,0], [0,-1]]

"A matrix is skew-symmetric if it is equal to the negative of its transpose. The matrix in option b) satisfies this property and is therefore skew-symmetric."

• ### Which of the following is not a property of the inverse of a matrix?

• a) The inverse of a matrix is unique
• b) If a matrix A is invertible, then its inverse is also invertible
• c) The product of a matrix and its inverse is equal to the identity matrix
• d) If a matrix A is not invertible, then its inverse can be obtained by taking the reciprocal of each element

Answer - d) The product of a matrix and its inverse is equal to the identity matrix

"If a matrix A is not invertible, then it does not have an inverse. Therefore, option d) is not a property of the inverse of a matrix."