CBSE Class 12th Mathematics Chapter 4 - Determinants Important Questions with Answers

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Find |AB|, if A = \(\left[\begin{array}{rr}
0 & -1 \\
0 & 2
\end{array}\right]\) and B = \(\left[\begin{array}{ll}
3 & 5 \\
0 & 0
\end{array}\right]\). (All India 2019)

Question 2.

If A = \(\left[\begin{array}{ll}
p & 2 \\
2 & p
\end{array}\right]\) and |A³| = 125 then find the value of p. (All India 2019)

Question 3.

If A is a square matrix satisfying A’A = I, write the value of |A|. (All India 2019)

Question 4.

If A and B are square matrices of the same order 3, such that |A| = 2 and AB = 27. Write the value of |B|. (Delhi 2019)

Question 5.

Using properties of determinants, show that (All India 2019)
\(\left|\begin{array}{ccc}
3 a & -a+b & -a+c \\
-b+a & 3 b & -b+c \\
-c+a & -c+b & 3 c
\end{array}\right|\) = 3(a + 6 + c) (ab + be + ca)

Question 6.

Using properties of determinants, prove the following (Delhi 2019)
\(\left|\begin{array}{ccc}
a+b+c & -c & -b \\
-c & a+b+c & -a \\
-b & -a & a+b+c
\end{array}\right|\) = 2(a + b) (b + c) (c + a)

Question 7.

Using properties of determinants, prove the following (Delhi 2019, 2012C, 2009)
\(\left|\begin{array}{ccc}
a & b & c \\
a-b & b-c & c-a \\
b+c & c+a & a+b
\end{array}\right|\) = a³ + b³ + c³ – 3abc

Question 8.

Show that for the matrix A = \(\left[\begin{array}{rrr}
1 & 1 & 1 \\
1 & 2 & -3 \\
2 & -1 & 3
\end{array}\right]\), A³ – 6A² + 5A + 11I = O. Hence, find A^-1. (All India 2019)

Question 9.

If A = \(\left[\begin{array}{lll}
1 & 3 & 4 \\
2 & 1 & 2 \\
5 & 1 & 1
\end{array}\right]\), find A^-1. Hence solve the system of equations
x + 3y + 4z = 8
2x + y + 2z = 5
and 5x + y + z = 7. (All India 2019)

Question 10.

If A = \(\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 0 & 2 \\
3 & 1 & 1
\end{array}\right]\), find A^-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12. (Delhi 2019)

Question 1.

Using properties of determinants, prove that (CBSE 2018)
\(\left|\begin{array}{ccc}
1 & 1 & 1+3 x \\
1+3 y & 1 & 1 \\
1 & 1+3 z & 1
\end{array}\right|\) = 9 (3xyz + xy + yz + zx).

Question 2.

Using properties of determinants, prove that (CBSE 2018C)
\(\left|\begin{array}{ccc}
5 a & -2 a+b & -2 a+c \\
-2 b+a & 5 b & -2 b+c \\
-2 c+a & -2 c+b & 5 c
\end{array}\right|\) = 12 (a + b + c) (ab + bc + ca).

Question 3.

Given A = \(\left[\begin{array}{cc}
2 & -3 \\
-4 & 7
\end{array}\right]\), compute A and show that 2A^-1 = 9I – A (CBSE 2018)

Question 4.

If A =\(\left[\begin{array}{ccc}
2 & -3 & 5 \\
3 & 2 & -4 \\
1 & 1 & -2
\end{array}\right]\), A^-1. Use it to solve the system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y – 2z = -3. (CBSE 2018)

Question 5.

Given A = \(\left[\begin{array}{lll}
5 & 0 & 4 \\
2 & 3 & 2 \\
1 & 2 & 1
\end{array}\right]\), B^-1 = \(\left[\begin{array}{lll}
1 & 3 & 3 \\
1 & 4 & 3 \\
1 & 3 & 4
\end{array}\right]\), compute (AB)^-1. (CBSE 2018C)

Question 6.

Using properties of determinants, prove that (Delhi 2017: All India 2017)
\(\left|\begin{array}{ccc}
x & x+y & x+2 y \\
x+2 y & x & x+y \\
x+y & x+2 y & x
\end{array}\right|\) = 9y²(x + y).

Question 7.

If for any 2 × 2 square matrix A, A(adj A) = \(\left[\begin{array}{ll}
8 & 0 \\
0 & 8
\end{array}\right]\), then write the value of |A|. (All India 2017)

Question 8.

Determine the product of \(\left[\begin{array}{ccc}
-4 & 4 & 4 \\
-7 & 1 & 3 \\
5 & -3 & -1
\end{array}\right]\) \(\left[\begin{array}{ccc}
1 & -1 & 1 \\
1 & -2 & -2 \\
2 & 1 & 3
\end{array}\right]\) and then Use to solve the system of equations
x – y + z = 4
x – 2y – 2z = 9
and 2x + y + 3z = 1. (All India 2017, Delhi 2012C)

Question 9.

Use Products \(\left[\begin{array}{rrr}
1 & -1 & 2 \\
0 & 2 & -3 \\
3 & -2 & 4
\end{array}\right]\left[\begin{array}{rrr}
-2 & 0 & 1 \\
9 & 2 & -3 \\
6 & 1 & -2
\end{array}\right]\)
to solve the system equations
x – y + 2z = 1
2y – 3z = 1
and 3x – 2y + 4z = 2. (Delhi 2017, Foreign 2011)