Rank of a Matrix

Q. If a, b, c are all different and $\left|\begin{array}{ccc}a& a^3& a^4-1\\ b& b^3& b^4-1\\ c& c^3& c^4-1\end{array}\right|=0,$ then the value of $abc(ab+bc+ca)$ is equal to:

1. $a-b-c$
2. $a-b+c$
3. $a+b+c$
4. $0$

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. Write number of non-zero matrices of order 2 x 3 with each entry 0, 1 or 2.

Q. While using rank method, what are the conditions for echelon form?

Q. If $A=\left[\begin{array}{ccc}3& 0& 0\\ 2& 2& 0\\ 4& 5& 3\end{array}\right],$ then which of the following is (are) true?

1. $A$ is invertible
2. $tr\left(adj\right(adj\left(A\right)\left)\right)=144$
3. $tr\left(adj\right(adj\left(A\right)\left)\right)=108$
4. $|adj\left(A\right)|$ is less than $400$

Q.

If matrix is in Row Echelon form then the Rank is equal to number of non zero rows.

1. True

2. False

Q.

If x, y, z are non-zero real numbers, then the inverse of matrix

$A=\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$ is

a) $\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$

b) $xyz\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$

c) $\frac{1}{xyz}\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$

d) $\frac{1}{xyz}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Q. If $A=\left[\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right],$ then the value of $\text{det(adj(adj}A\left)\right)$ is

1. ${14}^4$
2. ${14}^3$
3. ${14}^2$
4. ${14}^1$

Q. If $P=\left[\begin{array}{cc}5& -3\\ 111& 336\end{array}\right]$ and $det(-3P^{2013}+P^{2014})={\alpha }^{\alpha }{\beta }^2(1+\gamma +{\gamma }^2)$ where $\alpha ,\beta ,\gamma$ are natural numbers, then

1. $\alpha =2013$
2. $\beta =3$
3. $\gamma =10$
4. $\alpha =2014$

Q.

Examine the consistency of the system of equations

$x+y+z=1,2x+3y+2z=2,ax+ay+2az=4$

Q.

$\left[\begin{array}{ccc}3& -1& 2\\ -3& 1& 2\\ -6& 2& 4\end{array}\right]$What is the rank of the matrix.

1. 1

2. 2

3. 3

4. 4

Q.

Determine order and degree(if defined) of differential equation

Q. Examine the consistency of the system of equations$3x-y-2z=2,2y-z=-1,3x-5y=3$

Q. The rank of the matrix $\left[\begin{array}{ccc}3& -1& 2\\ -3& 1& 2\\ -6& 2& 4\end{array}\right]$ is

1. 1
2. 2
3. 3
4. 4

Q. If $A=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right],$ then $A^{50}$ is

1. $\left[\begin{array}{cc}1& 0\\ 0& 50\end{array}\right]$
2. $\left[\begin{array}{cc}1& 25\\ 0& 1\end{array}\right]$
3. None of these
4. $\left[\begin{array}{cc}1& 0\\ 50& 1\end{array}\right]$

Q.

If matrix is in Row Echelon form then the Rank is equal to number of non zero rows.

1. True

2. False

Q.

Evaluate

Q.

Which of the following is true if A and C are coefficient and augmented matrices respectively for a system of linear equation. Here n = number of unknowns

1. Rank A = Rank of C=n, Infinite solutions

2. Rank A = Rank of C = r, r < n, Infinite solutions

3. Rank A = Rank of C = n, unique solution

4. Rank A = Rank of C = r< n, unique solution

Q. The rank of the matrix $\left[\begin{array}{ccc}-1& 2& 5\\ 2& -4& a-4\\ 1& -2& a+1\end{array}\right]$ is if $a=1$ and if $a=-6$

1. 2
2. 1
3. 3
4. \N

Q.

Given A and C are coefficient and augmented matrices respectively for a system of linear equations. Which of the following cases tells if the equations are consistent?

1. Rank of A > Rank of C

2. Rank of A < Rank of C

3. Rank of A = Rank of C

4. None of these

Q.

For each of the differential equations given below, indicate its order and degree (if defined).

(i)

(ii)

(iii)

Q. The number of solution of the pair of equations 2sin$-cos2$=0 , 2cos$-3sin$=0 in the interval [0, 360] is _______ 1) 0 2) 1 3) 2 4) 4

Q. Examine the consistency of the system of equations$x+3y=5$, $2x+6y=8$

Q. Assertion :Consider the system of equation $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$. If the system has infinite number of solutions, then $\mu =10$.. Reason: The determinant $\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=0$ for $\mu =10$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. The rank of the matrix $\left[\begin{array}{ccc}-1& 2& 5\\ 2& -4& a-4\\ 1& -2& a+1\end{array}\right]$ is

1. 1 if a = 6
2. 2 if a = 1
3. 3 if a = 2
4. 1 if a = -6

Q.

Match the following

$x+y+z=2$

$x+y+\lambda z=1$

$x+\mu y+2y=3$

(a) $\mu =1,\lambda =1$ (i) No solution

(b) $\mu =2,\lambda =3$ (ii) Unique Solution

(c) $\mu =1,\lambda =0$ (iii) Infinite Solutions

1. a-ii; b – iii: c- ii

2. a-i; b – ii: c- iii

3. a-iii; b – iii: c- i

4. a-i; b – iii: c- ii

Q.

The Rank of the following matrix is

$\left[\begin{array}{cccc}1& 0& 0& 3\\ 0& 1& 1& 2\\ 0& 0& 0& 0\\ 0& 0& 1& 1\end{array}\right]-----------$

__

Q.

Which of the following is true if A and C are coefficient and augmented matrices respectively for a system of linear equation. Here n = number of unknowns

1. Rank A = Rank of C=n, Infinite solutions

2. Rank A = Rank of C = r, r < n, Infinite solutions

3. Rank A = Rank of C = n, unique solution

4. Rank A = Rank of C = r< n, unique solution

Q.

The solution set the given set of equations will be

$x+y+z=6$

$x+2y+3z=10$

$x+2y+z=1$

1. Unique trivial solutions

2. Unique non trivial solution

3. No Solutions

4. Infinite solutions

Q.

The rank of the matrix, $\begin{array}{c}A=\left[\begin{array}{cccc}2& 3& 1& 4\\ 0& 1& 2& -1\\ 0& -2& -4& 2\end{array}\right]is\end{array}$

1. 2

2. 3

3. 1

4. Indeterminate