Mathematics

Linear Algebra

449 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

if $\begin{bmatrix}2 &1 \ 7 &4 \end{bmatrix}$A$\begin{bmatrix}-3 &2 \ 5 &-3 \end{bmatrix}=\begin{bmatrix}1 &0 \ 0&1 \end{bmatrix}$, then matrix A equals

  1. $\begin{bmatrix}7 &5 \\ -11 &-8 \end{bmatrix}$
  2. $\begin{bmatrix}2 & 1 \\ 5 & 3 \end{bmatrix}$
  3. $\begin{bmatrix}7 & 34 \\ 1 & 5 \end{bmatrix}$
  4. $\begin{bmatrix}5 & 13 \\ 3 & 8 \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\begin{bmatrix}2 &1 \ 7 &4 \end{bmatrix}$A$\begin{bmatrix}-3 &2 \ 5 &-3 \end{bmatrix}=\begin{bmatrix}1 &0 \ 0&1 \end{bmatrix}$

$P=\begin{bmatrix}2 &1 \ 7 &4 \end{bmatrix},Q=\begin{bmatrix}-3 &2 \ 5 &-3 \end{bmatrix}, R=\begin{bmatrix}1 & 0 \ 0 & 1 \end{bmatrix}$

$PAQ = R \Rightarrow  A = P^{-1}RQ^{-1}$

$\Rightarrow A=P^{-1}Q^{-1}=(QP)^{-1}$

$QP=\begin{bmatrix}8 &5 \ -11 &-7 \end{bmatrix}$

$\therefore A=(QP)^{-1}=\begin{bmatrix}7 &5 \ -11 &-8 \end{bmatrix}$ 

Hence, option A.

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

Lets $A=\begin{bmatrix} 0&5 \-5 & 0\end{bmatrix}$ be a skew symmetric matrix and $I + A$ is non singular, then the matrix $B = (I - A)(I + A)^{-1}$ is

  1. an Orthogonal Matrix

  2. an Idempotent Matrix

  3. a Nilpotent Matrix

  4. Data Insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$B=(I-A)(I+A)^{-1}$

$B^{T}=[(I+A)^{-1}]^{T}(I-A)^{T}$
$\Rightarrow B^{T}=[(I+A)^{T}]^{-1}(I-A)^{T}$
$\Rightarrow B^{T}=(I-A)^{-1}(I+A)$             since $A^{T}=-A$
$B^{-1}=(I+A)(I-A)^{-1}$
In this case commutativity holds, so,
$B^{T}=B^{-1}\Rightarrow B\text{ is Orthogonal Matrix}$

Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

If $x,y,z$ are in A.P. then the value of the det A where $A=\begin{bmatrix} 4 & 5 & 6 & x \ 5 & 6 & 7 & y \ 6 & 7 & 8 & z \ x & y & z & 0 \end{bmatrix},$ is 

  1. $0$
  2. $1$
  3. $2$
  4. $-1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$|A|=\begin{vmatrix} 4 & 5 & 6 & x \\ 5 & 6 & 7 & y \\ 6 & 7 & 8 & z \\ x & y & z & 0 \end{vmatrix}$
$=-x\begin{vmatrix} 5 & 6 & x \\ 6 & 7 & y \\ 7 & 8 & z \end{vmatrix}+y\begin{vmatrix} 4 & 6 & x \\ 5 & 7 & y \\ 6 & 8 & z \end{vmatrix}-z\begin{vmatrix} 4 & 5 & x \\ 5 & 6 & y \\ 6 & 7 & z \end{vmatrix}+0\begin{vmatrix} 4 & 5 & 6 \\ 5 & 6 & 7 \\ 6 & 7 & 8 \end{vmatrix}$
$=-x(0)+y(0)-z(0)+0$
Determinate value of a matrix is 
zero if all of its rows or
column are in A.P. 
In all above $3\times 3$ determinate,
each column is A.P.
$\Rightarrow |A|=0$
Multiple choice

What are the eigenvalues of the matrix [[2, 1], [-1, 2]]?

  1. 1 and 3

  2. 2 and 4

  3. 1 and -1

  4. 2 and -2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Eigenvalues are the values of lambda for which the equation (A - lambda*I)x = 0 has a non-trivial solution. Solving this equation for the given matrix, we get the characteristic polynomial lambda^2 - 4*lambda + 5 = 0. The roots of this polynomial are 1 and 3, which are the eigenvalues of the matrix.

Multiple choice

Which of the following matrices is diagonalizable?

  1. [[2, 1], [-1, 2]]

  2. [[1, 2], [3, 4]]

  3. [[2, 3], [-3, 2]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A matrix is diagonalizable if it is similar to a diagonal matrix. This means that there exists an invertible matrix P such that P^-1*A*P is a diagonal matrix. The matrix [[2, 3], [-3, 2]] is diagonalizable because it has distinct eigenvalues and a complete set of eigenvectors.

Multiple choice

What is the determinant of the matrix [[1, 2], [3, 4]]?

  1. -2

  2. 2

  3. -4

  4. 4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of a matrix is a scalar value that can be calculated using various methods. For a 2x2 matrix, the determinant is given by the formula det([[a, b], [c, d]]) = ad - bc. In this case, det([[1, 2], [3, 4]]) = (1)(4) - (2)(3) = -2.

Multiple choice

Which of the following matrices is invertible?

  1. [[1, 2], [3, 6]]

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [0, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

An invertible matrix is a square matrix that has an inverse matrix. The inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. The matrix [[2, 1], [1, 2]] is invertible because its determinant is non-zero.

Multiple choice

What is the null space of the matrix [[1, 2], [3, 4]]?

  1. The set of all vectors x such that Ax = 0

  2. The set of all vectors x such that Ax = 1

  3. The set of all vectors x such that Ax = 2

  4. The set of all vectors x such that Ax = 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The null space of a matrix A is the set of all vectors x such that Ax = 0. This is also known as the kernel of the linear transformation represented by A.

Multiple choice

Which of the following is an orthogonal matrix?

  1. [[1, 0], [0, 1]]

  2. [[1, 2], [3, 4]]

  3. [[0, 1], [1, 0]]

  4. [[1, 2], [2, 1]]
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

An orthogonal matrix is a square matrix whose inverse is equal to its transpose. This means that the rows and columns of an orthogonal matrix are orthogonal to each other. The matrix [[1, 0], [0, 1]] is orthogonal because its inverse is also [[1, 0], [0, 1]].

Multiple choice

What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

  1. 1

  2. 2

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. The matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] has rank 2 because its first two rows are linearly independent, but its third row is a linear combination of the first two rows.

Multiple choice

Which of the following is a singular matrix?

  1. [[1, 2], [3, 4]]

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [0, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A singular matrix is a square matrix whose determinant is zero. The matrix [[0, 1], [0, 0]] is singular because its determinant is 0.

Multiple choice

What is the trace of the matrix [[1, 2], [3, 4]]?

  1. 5

  2. 7

  3. 9

  4. 11
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The trace of a matrix is the sum of its diagonal elements. The trace of the matrix [[1, 2], [3, 4]] is 1 + 4 = 5.

Multiple choice

Which of the following matrices is symmetric?

  1. [[1, 2], [3, 4]]

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [1, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A symmetric matrix is a square matrix that is equal to its transpose. The matrix [[2, 1], [1, 2]] is symmetric because its transpose is also [[2, 1], [1, 2]].

Multiple choice

What is the product of the matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]]?

  1. [[19, 22], [43, 50]]

  2. [[17, 20], [39, 46]]

  3. [[21, 24], [47, 54]]

  4. [[15, 18], [35, 42]]
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product of two matrices is calculated by multiplying the elements of the rows of the first matrix by the elements of the columns of the second matrix and adding the products. The product of the matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]] is [[19, 22], [43, 50]].

Multiple choice

Which of the following is an elementary matrix?

  1. [[1, 2], [3, 4]]

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [1, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An elementary matrix is a square matrix that can be obtained from the identity matrix by performing a single elementary row operation. The matrix [[0, 1], [1, 0]] is an elementary matrix because it can be obtained from the identity matrix by swapping the first and second rows.