Linear Algebra Multiple Choice Questions on “Types and Properties of Matrices”.
1. Find the transpose of the given Matrix.
(begin{bmatrix}1&3&-2\-1&7&0\1&0&8end{bmatrix})
a) (begin{bmatrix}1&-1&1\3&7&0\-2&0&8end{bmatrix})
b) (begin{bmatrix}1&1&1\3&7&0\-2&0&8end{bmatrix})
c) (begin{bmatrix}1&-1&1\3&7&22\-2&0&8end{bmatrix})
d) (begin{bmatrix}1&3&-2\-1&7&0\1&0&8end{bmatrix})
Answer: a
Explanation: We know that in a transpose matrix,
The rows and columns get interchanged.
So the transpose of the given matrix will be
(begin{bmatrix}1&-1&1\3&7&0\-2&0&8end{bmatrix})
.
2. Which of the following matrix is Skew Symmetric?
a) (begin{bmatrix}0&1\-1&0end{bmatrix})
b) (begin{bmatrix}0&1\1&0end{bmatrix})
c) (begin{bmatrix}0&3\-1&9end{bmatrix})
d) (begin{bmatrix}8&-2\-1&3end{bmatrix})
Answer: a
Explanation: We know for a Skew Symmetric Matrix,
A=-AT
So for option (begin{bmatrix}0&1\-1&0end{bmatrix}),
AT=(begin{bmatrix}0&-1\1&0end{bmatrix})=(-1)(begin{bmatrix}0&1\-1&0end{bmatrix})=A.
3. Which of the following matrix are singular?
a) (begin{bmatrix}31&12\26&8end{bmatrix})
b) (begin{bmatrix}1&11\2&8end{bmatrix})
c) (begin{bmatrix}13&12\22&8end{bmatrix})
d) (begin{bmatrix}3&12\2&8end{bmatrix})
Answer: d
Explanation: We know that,
A singular matrix is one whose determinant = 0
(begin{bmatrix}3&12\2&8end{bmatrix})
Determinant=24-24=0
Thus it is a singular matrix.
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