Mathematics

Linear Algebra

510 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 $AB=0$ for the matrices
$A=\left[ \begin{matrix} \cos ^{ 2 }{ \theta  }  & \cos { \theta  } \sin { \theta  }  \ \cos { \theta  } \sin { \theta  }  & \sin ^{ 2 }{ \theta  }  \end{matrix} \right] $ and $B=\left[ \begin{matrix} \cos ^{ 2 }{ \phi  }  & \cos { \phi  } \sin { \phi  }  \ \cos { \phi  } \sin { \phi  }  & \sin ^{ 2 }{ \phi  }  \end{matrix} \right] $ then $\theta-\phi $ is

  1. an odd multiple of $\dfrac{\pi}{2}$
  2. an odd multiple of ${\pi}$
  3. an odd even of $\dfrac{\pi}{2}$
  4. $0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$A=\begin{bmatrix} { \cos {  }  }^{ 2 }\theta  & \cos { \theta  } \sin { \theta  }  \\ \cos { \theta  } \sin { \theta  }  & { \sin {  }  }^{ 2 }\theta  \end{bmatrix}$      $B=\begin{bmatrix} { \cos {  }  }^{ 2 }  & \cos { \phi  } \sin { \phi  }  \\ \cos { \phi  } \sin { \phi  }  & { \sin {  }  }^{ 2 }\phi  \end{bmatrix}$

$AB = \begin{bmatrix} { \cos {  }  }^{ 2 }\theta  & \cos { \theta  } \sin { \theta  }  \\ \cos { \theta  } \sin { \theta  }  & { \sin {  }  }^{ 2 }\theta  \end{bmatrix}$  $ \begin{bmatrix} { \cos {  }  }^{ 2 }\phi  & \cos { \phi  } \sin { \phi  }  \\ \cos { \phi  } \sin { \phi  }  & { \sin {  }  }^{ 2 }\phi  \end{bmatrix}$

$=\begin{bmatrix} { \cos {  }  }^{ 2 }\theta \cos^{2}\phi + \cos { \theta  } \sin { \theta  } \cos\phi \sin\phi & {\cos^2\theta \cos\phi \sin\phi+\sin^2\phi \sin\theta \cos\theta} \\ \cos { \theta  } \sin { \theta  } \cos^{2}\phi  + { \sin {  }  }^{ 2 }\theta \cos\phi  \sin\phi  & \cos\theta \sin\theta \cos\phi  { \sin {  }  }^{ 2 }\phi  +\sin^{2}\theta\sin^{2}\phi   \end{bmatrix}$

$= \begin{bmatrix} 0  & 0  \\ 0  & 0  \end{bmatrix}$  

$\Rightarrow $ $\begin{bmatrix} { \cos {  }  }^{ 2 }\theta \cos \phi \cos (\theta -\phi)  & \cos { \theta  } \sin { \phi  } \cos (\theta-\phi) \\ \sin {\phi  } \cos {\phi   }  & { \sin {  }  }^{ 2 }\theta \sin \phi \cos (\theta -\phi)  \end{bmatrix}$ = $\begin{bmatrix} 0  & 0  \\ 0  & 0  \end{bmatrix}$

$\Rightarrow$ $\cos (\theta - \phi )$$\begin{bmatrix} { \cos {  }  }^{ 2 }\theta \cos \phi  & \cos { \theta  } \sin { \phi  }  \\ \sin {\phi  } \cos {\phi   }  & { \sin {  }  }^{ 2 }\theta \sin \phi   \end{bmatrix}$ = $\begin{bmatrix} 0  & 0  \\ 0  & 0  \end{bmatrix}$

$\Rightarrow \cos (\theta-\phi) (\cos \theta \cos \phi \sin \theta \sin \phi - \cos \theta \cos \phi \sin \theta \sin \phi ) = 0$

$\Rightarrow \cos (\theta - \phi) = 0$

$\theta - \phi = (2n+1)\dfrac{\pi}{2}$

i.e an odd multiple of $\dfrac{\pi}{2}$ 
Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices
Let $A$ be a matrix of order $2 \times 2$ such that $A^2 = 0$ then $A^2 - (a + d)A + (ad - bc) I$ is equal to
  1. $I$
  2. $0 _{2\times 2}$
  3. $-I$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

$A^2=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} a^2+bc & ab+bd \\ ca+cd & bc+d^2 \end{pmatrix}............................(1)$

$-(a+d)A=\begin{pmatrix} -a^2-ad & -ab-bd \\ -ac-cd & -ad-d^2 \end{pmatrix}..............(2)$

$(ad-bc)I=\begin{pmatrix} ad-bc & 0 \\ 0 & ad-bc \end{pmatrix}.............(3)$
Adding 1,2,3 we get,
$A^2-(a+d)A+(ad-bc)I=\begin{pmatrix} a^2+bc-a^2-ad+ad-bc & ab+bd-ab-bd \\ ac+cd-ac-cd & bc+d^2-ad-d^2+ad-bc \end{pmatrix}=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

Hence, the value of $A^2-(a+d)A+(ad-bc)I=0 _{2\times 2}$

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

Let $A$ and $B$ are two matrices such that $AB =BA$, then for every $n\in N$,

  1. $A^nB=BA^n$
  2. $(AB)^n = A^nB^n$
  3. $(A+B)^n=$ $^nC _0A^n+$ $^nC _1A^{n-1}B^1+$ $^nC _2A^{n-2}B^2+ ... + ^nC _n\:B^n$.
  4. $A^{2n}-B^{2n}=(A^n-B^n)(A^n+B^n)$
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

$A^2B =A(AB) =A(BA) =(AB)A$

$= (BA)A =BA^2$

Similarly, $A^3B=BA^3$

In general $A^nB=BA^n: \forall : n\geq 1$

(b) and (c) hold as $AB =BA$.

Also, $(A^n -B^n) (A^n + B^n)$

$=A^nA^n-B^n:A^n+A^n:B^n-B^n:B^n$

$=A^{2n}-B^{2n}$

Hence, options A,B,C and D.

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

If $D _1$ and $D _2$ are two $3\times 3$ diagonal matrices, then

  1. $D _1\:D _2$ is diagonal matrix
  2. $D _1\:D _2=D _2\:D _1$
  3. $D _1^2+D _2^2$ is a diagonal matrix

  4. none of these
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

Let ${ D } _{ 1 }=\begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix},{ D } _{ 2 }=\begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z \end{bmatrix}$

Then

${ D } _{ 1 }{ D } _{ 2 }=\begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}\begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z \end{bmatrix}=\begin{bmatrix} ax & 0 & 0 \ 0 & by & 0 \ 0 & 0 & cz \end{bmatrix}\$


$ { D } _{ 2 }{ D } _{ 1 }=\begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z \end{bmatrix}\begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}=\begin{bmatrix} xa & 0 & 0 \ 0 & yb & 0 \ 0 & 0 & zc \end{bmatrix}$

As $ax=xa,by=yb,cz=zc$

${ D } _{ 1 }{ D } _{ 2 }={ D } _{ 2 }{ D } _{ 1 }$

${ { D } _{ 1 } }^{ 2 }+{ { D } _{ 2 } }^{ 2 }=\begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}\begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}+\begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z \end{bmatrix}\begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z \end{bmatrix}\$

$ =\begin{bmatrix} { a }^{ 2 } & 0 & 0 \ 0 & { b }^{ 2 } & 0 \ 0 & 0 & { c }^{ 2 } \end{bmatrix}+\begin{bmatrix} { x }^{ 2 } & 0 & 0 \ 0 & { y }^{ 2 } & 0 \ 0 & 0 & { z }^{ 2 } \end{bmatrix}=\begin{bmatrix} { { a }^{ 2 }+x }^{ 2 } & 0 & 0 \ 0 & { { b }^{ 2 }+y }^{ 2 } & 0 \ 0 & 0 & { { c }^{ 2 }+z }^{ 2 } \end{bmatrix}$

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 is the primary application of linear algebra in signal processing?

  1. Solving systems of linear equations

  2. Matrix transformations

  3. Calculating eigenvalues and eigenvectors

  4. Decomposing signals into linear combinations of basis vectors
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Linear algebra is used in signal processing to decompose signals into linear combinations of basis vectors, which allows for efficient analysis, filtering, and compression of signals.

Multiple choice

In control theory, what is the state-space representation of a system based on?

  1. Systems of linear equations

  2. Matrix transformations

  3. Eigenvalues and eigenvectors

  4. All of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The state-space representation of a system in control theory is based on systems of linear equations that describe the system's dynamics. It allows for analysis and design of control systems using linear algebra techniques.

Multiple choice

What is the purpose of using numerical linear algebra in applications?

  1. Solving systems of linear equations

  2. Finding eigenvalues and eigenvectors

  3. Approximating integrals

  4. Computing matrix inverses
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Numerical linear algebra is used to solve various problems involving matrices and vectors, including solving systems of linear equations, finding eigenvalues and eigenvectors, approximating integrals, and computing matrix inverses.

Multiple choice

Which numerical method is commonly used for finding the eigenvalues and eigenvectors of a matrix?

  1. Power method

  2. QR algorithm

  3. Jacobi method

  4. Gauss-Seidel method
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The QR algorithm is a widely used numerical method for finding the eigenvalues and eigenvectors of a matrix by iteratively transforming the matrix into upper triangular form.

Multiple choice

What is the matrix representation of the linear transformation f(x) = 2x - 1?

  1. [[2, 0], [0, -1]]

  2. [[2, 1], [0, -1]]

  3. [[2, -1], [0, 1]]

  4. [[2, 0], [1, -1]]
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The matrix representation of a linear transformation is a matrix whose columns are the coordinates of the transformed basis vectors. In this case, the basis vector is [1, 0], and its transformed vector is [2, -1]. Therefore, the matrix representation is [[2, -1]].

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.