Determinants: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Find $x$ if $\left|\begin{array}{cc}x& 4\\ 1& x\end{array}\right|=0$.
2. Write adj $\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$.
3. Evaluate $\left|\begin{array}{ccc}1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\end{array}\right|$.
4. If $A^2=I$, what is $A^{-1}$?
5. Area with vertices $(0,0),(4,0),(0,3)$.
6. For which $k$ is $\left[\begin{array}{cc}k& 2\\ 3& 6\end{array}\right]$ singular?
7. $|I_3|=$?
8. If $|A|=-3$ (order 2), $|\mathrm{adj}A|=$?
9. Minor $M_{21}$ of $\left|\begin{array}{cc}1& 2\\ 3& 4\end{array}\right|$.
10. $|AB|$ if $|A|=2$, $|B|=-3$.

D. Short answer I (2 marks each)

1. Find the inverse of $\left[\begin{array}{cc}3& -1\\ 2& 1\end{array}\right]$.
2. Find $k$ if the area of the triangle $(1,1),(2,3),(k,0)$ is 2.
3. Using determinants, find the line through $(1,1)$ and $(3,7)$.
4. Show that $\left|\begin{array}{ccc}a& b& c\\ a& b& c\\ 1& 2& 3\end{array}\right|=0$ without expanding.
5. If A is $3\times 3$ with $|A|=3$, find $|2\mathrm{adj}A|$.
6. Examine the consistency of $x+y=3$, $2x+2y=7$.
7. If $A=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$, verify $A^2-4A+I=O$ and find $A^{-1}$.
8. Find $x$ if $\left|\begin{array}{ccc}x& 2& 3\\ 0& 1& 4\\ 0& 0& x\end{array}\right|=9$.

E. Short answer II (3 marks each)

1. Find adj A and $A^{-1}$ for $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$.
2. Solve $2x+5y=1$, $3x+2y=7$ by the matrix method.
3. Show that the points $(a,0)$, $(0,b)$, $(1,1)$ are collinear if $\frac{1}{a}+\frac{1}{b}=1$.
4. Find the values of $k$ for which $kx+2y=5$, $3x+y=1$ has a unique solution.
5. Evaluate $\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|$.
6. If $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$, $B=\left[\begin{array}{cc}2& 0\\ 1& 1\end{array}\right]$, verify $(AB{)}^{-1}=B^{-1}{A}^{-1}$.

F. Long answer (5 marks each)

1. Using matrices, solve $x+y+z=6$, $y+3z=11$, $x-2y+z=0$.
2. Find $A^{-1}$ for $A=\left[\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right]$ and hence solve $x+2y-z=-1$, $-x+y+2z=3$, $2x-y+z=4$.

G. HOTS (higher-order thinking)

1. If $A^2-A+I=O$, find $A^{-1}$ in terms of A.
2. Can a $2\times 2$ matrix with $|A|=0$ have $\mathrm{adj}A=O$?
3. If every entry of a $3\times 3$ matrix is doubled, how does $|\mathrm{adj}A|$ change?
4. Why is the determinant of an odd-order skew-symmetric matrix 0?

20 MCQs

1. $\left|\begin{array}{cc}\sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{array}\right|$
   (a) 0 (b) $-1$ (c) 1 (d) $\sin 2\theta$
2. $|3A|$ for $2\times 2$, $|A|=4$
   (a) 36 (b) 12 (c) 108 (d) 24
3. $|\mathrm{adj}A|$, $3\times 3$, $|A|=-2$
   (a) $-2$ (b) 8 (c) $-8$ (d) 4
4. A is invertible iff
   (a) A is symmetric (b) $|A|\ne 0$ (c) $|A|=1$ (d) A is diagonal
5. $|A^{-1}|$ if $|A|=\frac{1}{2}$
   (a) $\frac{1}{2}$ (b) 2 (c) $\frac{1}{4}$ (d) 4
6. Cofactor $A_{21}$ of $\left|\begin{array}{cc}5& 7\\ 2& 3\end{array}\right|$
   (a) 7 (b) $-2$ (c) 2 (d) $-7$
7. If $|A|=0$ and $(\mathrm{adj}A)B\ne O$, the system is
   (a) inconsistent (b) unique (c) infinite (d) trivial
8. $\left|\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 5& 7& 9\end{array}\right|$
   (a) 6 (b) 1 (c) 0 (d) $-6$
9. Area of triangle $(0,0),(2,0),(0,2)$
   (a) 4 (b) 8 (c) 1 (d) 2
10. $A^{-1}$ of $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
    (a) $-A$ (b) $A$ (c) $I$ (d) $O$
11. $|AB|$ if $|A|=3$, $|B|=4$
    (a) 7 (b) 1 (c) 12 (d) 34
12. $x$ if $\left|\begin{array}{cc}x& 1\\ 9& x\end{array}\right|=0$
    (a) $\pm 3$ (b) 3 (c) 9 (d) $\pm 9$
13. $\mathrm{adj}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$
    (a) $\left[\begin{array}{cc}4& 2\\ 3& 1\end{array}\right]$ (b) $\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$ (c) $\left[\begin{array}{cc}1& -2\\ -3& 4\end{array}\right]$ (d) $\left[\begin{array}{cc}-4& 2\\ 3& -1\end{array}\right]$
14. $|A\mathrm{adj}A|$, $3\times 3$, $|A|=2$
    (a) 2 (b) 4 (c) 8 (d) 16
15. $(kA{)}^{-1}$
    (a) $k{A}^{-1}$ (b) $A^{-1}$ (c) $k^n{A}^{-1}$ (d) $\frac{1}{k}{A}^{-1}$
16. Determinant of $3\times 3$ skew-symmetric
    (a) 0 (b) 1 (c) $-1$ (d) any
17. Sum $a_{11}{A}_{21}+a_{12}{A}_{22}+a_{13}{A}_{23}$
    (a) 0 (b) $|A|$ (c) 1 (d) $-|A|$
18. If $A=\mathrm{diag}(2,3,4)$, $|A|=$
    (a) 9 (b) 6 (c) 0 (d) 24
19. $\det (A^{-1})$ for $2\times 2$
    (a) $\det A$ (b) $\frac{1}{\det A}$ (c) 1 (d) 0
20. $\left[\begin{array}{cc}2& k\\ 4& 6\end{array}\right]$ singular when $k=$
    (a) 2 (b) 4 (c) 3 (d) 6

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: If A is $3\times 3$ with $|A|=2$, then $|2A|=16$. R: $|kA|=k^n|A|$.
2. A: $\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$ has no inverse. R: its determinant is 0.
3. A: The points $(1,1),(2,2),(3,3)$ form a triangle of area 1. R: area $=\frac{1}{2}|\Delta |$.
4. A: $|\mathrm{adj}A|=|A{|}^2$ for every $3\times 3$ A. R: $A(\mathrm{adj}A)=|A|I$.
5. A: If $|A|\ne 0$, $AX=B$ has exactly one solution. R: then $X=A^{-1}B$.
6. A: $(A+B{)}^{-1}=A^{-1}+B^{-1}$. R: $(AB{)}^{-1}=B^{-1}{A}^{-1}$.

Key: 1 (a) · 2 (a) · 3 (d) · 4 (a) · 5 (a) · 6 (d)

Answers to the practice set