Cofactor

Q. If A = $\left|\begin{array}{ccc}5& 6& 3\\ -4& 3& 2\\ -4& -7& 3\end{array}\right|,$ then cofactors of the elements of $2^{nd}$ row are

1. 39, -3, 11
2. -39, 3, 11
3. -39, -3, 11
4. -39, 27, 11

Q.

How do you evaluate a matrix.

Q.

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3-2x^2+3x-4=0$, then the value of ${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2$

1. $-7$

2. $-5$

3. $-3$

4. $0$

Q. Let $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right],x\in R$ and $A^4=\left[a_{ij}\right]$. If $a_{11}=109$, then $a_{22}$ is equal to

Q.

The minors of - 4 and 9 and the co-factors of - 4 and 9 in determinant $\left|\begin{array}{ccc}-1& -2& 3\\ -4& -5& -6\\ -7& 8& 9\end{array}\right|$ are respectively

1. 42, 3 ; – 42, 3

2. –42, –3 ; 42, –3

3. 42, 3 ; – 42, – 3

4. 42, 3; 42, 3

Q. If $A=\left[\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right]$. Write the cofactor of the element a₃₂.

Q. The cofactor of the element '4' in the determinant $\left|\begin{array}{cccc}1& 3& 5& 1\\ 2& 3& 4& 2\\ 8& 0& 1& 1\\ 0& 2& 1& 1\end{array}\right|$ is

1. 4
2. -10
3. 10
4. -4

Q. The cofactor of element 5 in the matrix $\left(\begin{array}{ccc}1& -3& 2\\ 4& -1& 2\\ 3& 5& 2\end{array}\right]$ will be .

1. -12
2. 14
3. 6
4. -30

Q. let a= aij is 3 x 3 scalar matrix such that a11+ a22+ a33 = 15. Then write matrix A

Q. The number of all possible square matrices of order $2$ formed by the elements $1,2$ and $3$ is ______

Q.

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

(A) 27

(B) 18

(C) 81

(D) 512

Q. In the matrix
$A=\left[\begin{array}{cccc}2& 5& 19& -7\\ 35& -2& \frac{5}{2}& 12\\ \sqrt{3}& 1& -5& 17\end{array}\right]$

$\left(i\right).$ Write the order of the matrix
$\left(ii\right).$ Write the numbers of element.
$\left(iii\right).$ Write the elements $a_{13},a_{21},a_{33},a_{24},a_{23}$

Q. Using Cofactors of elements of third column, evaluate

Q.

If A1, B1, C1 … are the cofactors of the elements a1, b1, c1... of the matrix

$\left[\begin{array}{ccc}a1& b1& c1\\ a2& b2& c2\\ a3& b3& c3\end{array}\right]$ then $\left|\begin{array}{cc}B2& C2\\ B3& C3\end{array}\right|=$

1. a1 D

2. (a1 +a2)D

3. a1a3D

4. None of these

Q.

What is the value of $c$ in $x^2+8x+c$ that would make it a perfect square trinomial?

Q. If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 2015& 2016& -1\end{array}\right],$ then $(2A^2+4A^4+6A^6)$ is equal to
$($Here, $I_3$ is unit matrix of order $3)$

1. $3I_3$
2. $12I_3$
3. $12A$
4. $6A^6$

Q. If A is a matrix of order 3x3 and |a| = 2 then the determinant of cofactor matrix of A is

Q.

If A1, B1, C1 … are the cofactors of the elements a1, b1, c1... of the matrix

$\left[\begin{array}{ccc}a1& b1& c1\\ a2& b2& c2\\ a3& b3& c3\end{array}\right]$ then $\left|\begin{array}{cc}B2& C2\\ B3& C3\end{array}\right|=$

1. a1 D

2. a1a3D

3. (a1 +a2)D

4. None of these

Q. List all the elements of the following set:

(iii) $C=\left\{\begin{array}{c}x:x\text{is an integer},-\frac{1}{2}<x<\frac{9}{2}\end{array}\right\}$

Q.

Write minors and cofactors of elements of following determinants

$\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|$

Q. If $f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& (x+1)x\\ 3x(x-1)& x(x-1)(x-2)& (x+1)x(x-1)\end{array}\right|$,

Then, $f\left(100\right)$ is equal to

1. $0$ (Zero)
2. $1$
3. $10$
4. $100$

Q. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
(a) 27
(b) 18
(c) 81
(d) 512

Q. If $\left|\begin{array}{ccc}x+a& a^2& a^3\\ x+b& b^2& b^3\\ x+c& c^2& c^3\end{array}\right|=0$ and $a\ne b\ne c,$ then the value of $x$ is:

1. $\frac{-abc}{ab+bc+ca}$
2. $\frac{abc}{(a-b)(b-c)(c-a)}$
3. $\frac{abc}{ab+bc+ca}$
4. $\frac{(a-b)(b-c)(c-a)}{abc}$

Q. The value of the determinant $\left|\begin{array}{ccc}1& 1& 1\\ {}^n{C}_{n-1}& {}^{n+1}{C}_n& {}^{n+2}{C}_1\\ {}^n{C}_{n-2}& {}^{n+1}{C}_{n-1}& {}^{n+2}{C}_2\end{array}\right|$

1. $2$
2. $1$
3. $0$
4. ${}^n{C}_{n-1}$

Q. The number of possible matrices of order 3 × 3 with each entry 2 or 0 is

(a) 9
(b) 27
(c) 81
(d) none of these

Q.

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

(A) 27

(B) 18

(C) 81

(D) 512

Q. If A = $\left|\begin{array}{ccc}5& 6& 3\\ -4& 3& 2\\ -4& -7& 3\end{array}\right|,$ then cofactors of the elements of $2^{nd}$ row are

1. -39, -3, 11
2. 39, -3, 11
3. -39, 3, 11
4. -39, 27, 11

Q. The number of all possible matrices of order 2 x 3with each entry 1 or-1 is

Q. Which of the following statement are false? (A) the rank of the unit matrix of order n is unity . (B) the rank of null matrix is zero. (C) the rank of m×n matrix cannot exceed the minimum of m and n . (D) equivalent matrix have the same rank

Q. If $△=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$ and $A_2,B_2,C_2$ are respectively cofactors of $a_2,b_2,c_2$ then $a_1{A}_2+b_1{B}_2+c_1{C}_2$ is equal to ?

1. $-△$
2. $0$
3. $△$
4. $noneofthese$