True False[1 Marks ]

Question 11 Mark

Which of the following statements are True or False.
If A, B and C are square matrices of same order, then AB = AC always implies that B = C.

Answer

False.Solution:
If $AB = AC$
$\Rightarrow B = C$ only if $A^{-1}$ exits.
As then $A^{-1}AB = A^{-1}AC$
$⇒ IB = IC$
$⇒ B = C$

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Question 21 Mark

Which of the following statements are True or False.
If A and B are two square matrices of the same order, then AB = BA.

Answer

False.Solution:
For two square matrices of same order it is not always true that AB = BA.

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Question 31 Mark

Which of the following statements are True or False.
If A and B are two square matrices of the same order, then A + B = B + A.

Answer

True.Solution:
Since, matrix addition is commutative ie, A + B = B + A, where A and B are two square matrices.

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Question 41 Mark

Which of the following statements are True or False.
Two matrices are equal if they have same number of rows and same number of columns.

Answer

False.Solution:
If two matrices have same number of rows and same number of columns, we cannot say two matrices are equal as their corresponding elements may be different.

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Question 51 Mark

Which of the following statements are True or False.If $\text{A}=\begin{bmatrix}2&3&-1\\1&4&2\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&3\\4&5\\2&1\end{bmatrix},$ then AB and BA are defined and equal.

Answer

False.Solution:
A and B are matrices of the order 2 × 3 and 3 × 2 respectively.
Now, the order of AB and BA is given by 2 × 2 and 3 × 3 respectively.
$\therefore\ \text{AB}\neq\text{BA}$

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Question 61 Mark

Which of the following statements are True or False.
If A is skew symmetric matrix, then $A^2$ is a symmetric matrix.

Answer

True.Solution:
$ \because\left[A^2\right]^{\prime}=\left[A^{\prime}\right]^2 $
$ =[-A]^2\left[\because A^{\prime}=-A\right] $
$ =A^2$
$\text { Hence, } A^2 \text { is symmetric matrix. }$

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Question 71 Mark

Which of the following statements are True or False.
Matrices of different order can not be subtracted.

Answer

True.Solution:
Two matrices of same order can be subtracted.

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Question 81 Mark

Which of the following statements are True or False.
If (AB)′ = B′ A′, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.

Answer

True.Solution:
Let A is of order m × n and B is of order p × q.
Therefore, the order of A’ and B’ will be given by n × m and q × p respectively.
Now, we know that, AB will be defined if and only if n = p.
The order of AB will be given by m × q.
Again, B'' will be defined if and only if p = n.
The order of B'A' will be given by q × m.
Hence, if (AB)' = B'A', where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.

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Question 91 Mark

Which of the following statements are True or False.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

Answer

True.Solution:
Let A, B and C are three matrices of same order
$\therefore$ A' = A, B' = B and C' = C
$\therefore$ (A + B + C)' = A' + B' + C' = A + B + C

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Question 101 Mark

Which of the following statements are True or False.
AA′ is always a symmetric matrix for any matrix A.

Answer

True.Solution:
Let us suppose A be a matrix of any order.
$\therefore$ (AA')' = (A')'A' = AA'

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Question 111 Mark

Which of the following statements are True or False.
If matrix AB = 0, then A = 0 or B = 0 or both A and B are null matrices.

Answer

False.Solution:
Since, for two non-zero matrices A and B of same order, it can be possible that A.B = 0 = null matrix.

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Question 121 Mark

Which of the following statements are True or False.
$(AB)^{-1}= A^{-1}. B^{-1}$, where A and B are invertible matrices satisfying commutative property with respect to multiplication.

Answer

True.Solution:
We have $(A B)\left(A^{-1} B^{-1}\right)$
$=(B A)\left(A^{-1} B^{-1}\right)$ (as it is given that $A B=B A$ )
$=B\left(A A^{-1}\right) B^{-1}$
$=B.I.B^{-1}=B.B^{-1}=1$
So, $(A B)^{-1}=A^{-1} B^{-1}$

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Question 131 Mark

Which of the following statements are True or False.
Matrix addition is associative as well as commutative.

Answer

True.Solution:
Matrix addition is associative as well as commutative i.e.,
(A + B) + C = A + (B + C) and A + B = B + A, where A, B and C are matrices of same order.

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Question 141 Mark

Which of the following statements are True or False.
Matrices of any order can be added.

Answer

False.Solution:
The necessary condition for addition of two matrices is that both the matrices are of the same order.

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Question 151 Mark

Which of the following statements are True or False.
A matrix denotes a number.

Answer

False.Solution:
A matrix is an ordered rectangular array of numbers of function.

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Question 161 Mark

Which of the following statements are True or False.
A square matrix where every element is unity is called an identity matrix.

Answer

False.Solution:
An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.

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Question 171 Mark

Which of the following statements are True or False.
If A and B are any two matrices of the same order, then (AB)′ = A′B′.

Answer

False.Solution:
$\because$ (AB)' = B'A'

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Question 181 Mark

Which of the following statements are True or False.
Matrix multiplication is commutative.

Answer

False.Solution:
If AB is defined, it is not necessary that BA is defined.
Also if AB and BA are defined, it not necessary that they have same order. Further if AB and BA are defined and have same order, it is not necessary their corresponding elements are equal.
So, in general AB^BA

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Question 191 Mark

Which of the following statements are True or False.
Transpose of a column matrix is a column matrix.

Answer

False.Solution:
We can obtain the transpose of a matrix by interchanging the rows and columns.

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Question 201 Mark

Which of the following statements are True or False.
If A and B are two matrices of the same order, then A - B = B - A.

Answer

False.Solution:
$\text{A}-\text{B}=-(\text{B}-\text{A})$
Thus $\text{A}-\text{B}\neq\text{B}-\text{A}$
However when $\text{A}-\text{B}=\text{B}-\text{A}$
$\text{A}-\text{B}=0$ or $\text{A}=\text{B}$

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Maths True False[1 Marks ]

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