Minor and Cofactor of a Matrix

Q.

If $\Delta =\left|\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right|$, then the co-factor of the element $a_{23}$ is

1. $5$
2. $-7$
3. $8$
4. $1$

Q.

What do you mean by matrix?

Q.

Let $a_n$ be the $n^{th}$ term of a G.P. of positive terms. If $\sum _{n=1}^{100}{a}_{2n+1}=200$ and $\sum _{n=1}^{100}{a}_{2n}=100$ then value of $\begin{array}{l}\sum _{n=1}^{200}{a}_n\end{array}$ is

1. $300$

2. $175$

3. $225$

4. $150$

Q.

If $A=\left[\begin{array}{ccc}1& -5& 7\\ 0& 7& 9\\ 11& 8& 9\end{array}\right]$ then the trace of the matrix is

Q.

Minor $M_{33}$ $($Minor of the element of $i^{\text{th}}$ row and $j^{\text{th}}$ column$)$ of the determinant $\left|\begin{array}{ccc}2& 3& 5\\ 2& -1& 8\\ 1& 2& 4\end{array}\right|$ is

1. $1$
2. $-32$
3. $-15$
4. $-8$

Q. If $A$ is a matrix of order $3\times 3,$ then the number of minors in determinant of $A$ are

Q. If $A=\left[a_{ij}\right]$ is a $2\times 2$ matrix. Then which of the following statements is/are always true?

1. Sum of minors of elements of $A$ is equal to sum of its elements
2. Sum of minors of elements of$A$ is equal to twice the sum of its elements
3. Product of minors of elements of $A$ is equal to the product of its elements
4. Product of minors of elements of $A$ is equal to the product of square of its elements

Q. If $A=\left[a_{ij}\right]$ is a $2\times 2$ matrix, such that sum of co-factors of its elements is equal to the sum of elements of $A.$ Then which of the following can be matrix $A$ ?

1. $\left[\begin{array}{cc}-1& 3\\ 0& 1\end{array}\right]$
2. $\left[\begin{array}{cc}1& -4\\ 4& 2\end{array}\right]$
3. $\left[\begin{array}{cc}3& 7\\ -7& -7\end{array}\right]$
4. $\left[\begin{array}{cc}-7& -4\\ 5& 7\end{array}\right]$

Q. Let a matrix $A=\left[\begin{array}{cc}\cos x& \sin x\\ \tan x& \cot x\end{array}\right],$ then which of the following statement(s) is(are) true for atleast one value of $x\in [0,\frac{\pi }{2}]$ ?
(where $C_{ij}$ is co-factor of element $\left[a_{ij}\right]$)

1. $C_{11}=1$
2. $C_{12}=1$
3. $C_{21}=1$
4. $C_{22}=1$

Q. Consider the matrix $A=\left[\begin{array}{ccc}2& 6& 8\\ 4& 5& 1\\ 3& 7& 9\end{array}\right].$ The cofactor of element $5$ in matrix $A$ is

[1 mark]

1. $6$
2. $-6$
3. $4$
4. $-4$

Q. Consider the matrix $A=\left[\begin{array}{ccc}1& 4& 3\\ 7& 6& 9\\ 2& 5& 8\end{array}\right].$ Then cofactor of element $9$ in matrix $A$ is

[2 marks]

1. $-3$
2. $3$
3. $5$
4. $-5$

Q. Let a matrix $A=\left[\begin{array}{cc}2& 3\sin x\\ 4\cos x& -1\end{array}\right],x\in R,$ then the maximum value of sum of minors of elements of $A$ is

Q. Consider the matrix $A=\left[\begin{array}{cc}3& 2\sin x\\ 1& -2\end{array}\right],$ where $x\in R.$ Then the maximum value of sum of minors of all elements of $A$ is

[2 marks]

1. $0$
2. $6$
3. $4$
4. $3$

Q.

Minor $M_{33}$ $($Minor of the element of $i^{\text{th}}$ row and $j^{\text{th}}$ column$)$ of the determinant $\left|\begin{array}{ccc}2& 3& 5\\ 2& -1& 8\\ 1& 2& 4\end{array}\right|$ is

1. $-32$
2. $-8$
3. $-15$
4. $1$

Q. If $\Delta =\left|\begin{array}{ccc}5& 3& 8\\ 2& 8& 1\\ 1& 2& 3\end{array}\right|,$ then minor of the element in $3^{\text{rd}}$ row and $2^{\text{nd}}$ column is

[1 mark]

1. $11$
2. $-11$
3. $7$
4. $-6$

Q.

If $\Delta =\left|\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right|$, then the co-factor of the element $a_{23}$ is

1. $1$
2. $5$
3. $-7$
4. $8$

Q. If $A=\left[a_{ij}\right]$ is a $2\times 2$ matrix. Then which of the following statement is always true :

1. Sum of co-factors of elements of $A$ is equal to sum of its elements
2. Product of co-factors of elements of $A$ is equal to the product of its elements
3. Product of co-factors of elements of $A$ is equal to the product of square of its elements
4. Sum of co-factors of elements of $A$ is equal to the difference in sum of its row elements