Tag: inverse of a matrix and linear equations

Questions Related to inverse of a matrix and linear equations

If the matrix $\begin{bmatrix} \alpha  & 2 & 2 \ -3 & 0 & 4 \ 1 & -1 & 1 \end{bmatrix}$ is not invertible, then:

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. $\alpha =-5$

  2. $\alpha =5$

  3. $\alpha =-0$

  4. $\alpha =-1$

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Correct Option: A
Explanation:

For a matrix to be not invertible, determinant of the matrix should be $0$
So, $\left| \begin{matrix} \alpha  & 2 & 2 \ -3 & 0 & 4 \ 1 & -1 & 1 \end{matrix} \right| =0$

$\Rightarrow -(-3)(2-(-2))-4(-\alpha -2)=0\ \Rightarrow 12+4\alpha +8=0\ \Rightarrow 4\alpha =-20\ \Rightarrow \alpha =-5$
Hence, option A is correct

Consider the following statements:
1. The matrix
               $\begin{pmatrix} 1 & 2 & 1 \ a & 2a & 1 \ b & 2b & 1 \end{pmatrix}$ is singular.
2. The matrix
              $\begin{pmatrix} c & 2c & 1 \ a & 2a & 1 \ b & 2b & 1 \end{pmatrix}$ is non-singular.
Which of the above statements is/are correct?

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

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Correct Option: A
Explanation:

For matrix

  $\begin{vmatrix} 1 & 2 & 1 \ a & 2a & 1 \ b & 2b & 1 \end{vmatrix}$


$C _2\rightarrow C _2-2C _1 $

$\begin{vmatrix} 1 & 0 & 1 \\ a & 0 & 1 \\ b & 0 & 1 \end{vmatrix}$ The determinant is $0$. So the matrix is singular.

For matrix,
$ \begin{vmatrix} c & 2c & 1 \\ a & 2a & 1 \\ b & 2b & 1 \end{vmatrix}$

$C _2\rightarrow C _2-2C _1 $

$\begin{vmatrix} c & 0 & 1 \\ a & 0 & 1 \\ b & 0 & 1 \end{vmatrix}$ The determinant is $0$. So the matrix is singular.

Thus, only $1$ is true.
Hence, option A is correct.

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. If $det(A)=\pm 1$, then ${A}^{-1}$ exists but all its entries are not necessarily integers.

  2. If $det(A)=\pm 1$, then ${A}^{-1}$ exists and all its entries are non integers

  3. If $det(A)=\pm 1$, then ${A}^{-1}$ exists and all its entries are integers

  4. If $det(A)=\pm 1$, then ${A}^{-1}$ need not exist

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Correct Option: C

If $A$ and $B$ are two non-zero square matrices of the same order such that the product $AB=0$, then

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. both $A$ and $B$ must be singular

  2. exactly one of them must be singular

  3. both of them are non singular

  4. none of these

Show answer
Correct Option: D
Explanation:

$AB=O$
taking determinant on both sides
$|AB|=|A||B|=0$
$\Rightarrow$ Either $A$ or $B$ should be a singular matrix.
Hence, option D.

Let $A=\begin{bmatrix} a & b\ c & d\end{bmatrix}$ be a $2\times 2$ matrix, where a, b, c and d take the values $0$ or $1$ only. The number of such matrices which have inverses is?

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. $8$

  2. $7$

  3. $6$

  4. $5$

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Correct Option: A
Explanation:

${ A }^{ -1 }=\frac { adj\quad A }{ \left| A \right|  } $

so $\left| A \right| \neq 0\ ad-bc\neq 0$
for  $ ad-bc\neq 0$ 
either $ad=1 ,bc=0$ or $ad=0 ,bc=1$
$ad=1 ,bc=0$ can be chosen in 4 ways 
similarly $ad=0 ,bc=1$ can be chosen in 4 ways
so number of matrices  formed$=4+4=8$

If $A$ is a nonsingular matrix satisfying $AB=BA+A$ then

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. $\left|B\right|=\left|I+B\right|$

  2. $\left|B\right|=\left|2I+B\right|$

  3. $\left|B\right|=\left|B-I\right|$

  4. $\left|B\right|=\left|B-2I\right|$

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Correct Option: A

If $A$ and $B$ and square matrix of the same order such that $AB=A$ and $BA=B$, then $A$ and $B$ are both:

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. Singular

  2. Non-singular

  3. Idempotent

  4. Involutory

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Correct Option: A

The number of $3\times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$ is 

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. Less than $4$

  2. $5$

  3. $6$

  4. At least $7$

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Correct Option: D

If A and B are two non-singular square matrices and AB=I, then which of the following is true ?

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. $BA = I$

  2. ${ A }^{ -1 }=B$

  3. ${ B }^{ -1 }=A$

  4. ${ A }^{ 2 }=B$

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Correct Option: A

If $A$ and $B$ are non-singular matrices, then _____

business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

  1. $(AB)^{-1} = A^{-1}B^{-1}$

  2. $AB = BA$

  3. $(AB)^T = A^T. B^T$

  4. $(AB)^{-1} = B^{-1} A^{-1}$

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Correct Option: A