Properties of determinants class 12

Concept 1: Evaluation of \(3 \times 3\) Determinants

A determinant of order 3 is a scalar value associated with a \(3 \times 3\) square matrix. It can be evaluated by expanding it along any of its three rows or three columns.

1. Expansion Method

To expand along the first row (\(R_1\)):

\[ \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1 \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} – b_1 \begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1 \begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix} \]

2. Sign Convention (Cofactor Signs)

The sign assigned to each element \(a_{ij}\) is \((-1)^{i+j}\). For a \(3 \times 3\) matrix, the signs are:

\[ \begin{bmatrix} + & – & + \\ – & + & – \\ + & – & + \end{bmatrix} \]

Solved Examples (Concept 1)

Example 1:

Evaluate \(\Delta = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}\).

Solution: Expanding along \(C_3\) (Column 3) is easiest due to the zeros: \(\Delta = 4 \begin{vmatrix} -1 & 3 \\ 4 & 1 \end{vmatrix} – 0 + 0 = 4[(-1)(1) – (4)(3)] = 4[-1 – 12] = 4(-13) = -52\).

Example 2:

Evaluate \(\Delta = \begin{vmatrix} 0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0 \end{vmatrix}\).

Solution: Expanding along \(R_1\): \(\Delta = 0 – \sin \alpha (0 – \cos \alpha \sin \beta) + (-\cos \alpha)(\sin \alpha \sin \beta – 0)\) \(\Delta = \sin \alpha \cos \alpha \sin \beta – \cos \alpha \sin \alpha \sin \beta = 0\).

Example 3:

Find values of \(x\) if \(\begin{vmatrix} x & 3 & 7 \\ 2 & x & -2 \\ 7 & 8 & x \end{vmatrix}\) has a value of 0 when \(x=5\) is one root.

Solution: Substitute \(x=5\): \(\begin{vmatrix} 5 & 3 & 7 \\ 2 & 5 & -2 \\ 7 & 8 & 5 \end{vmatrix} = 5(25+16) – 3(10+14) + 7(16-35) = 5(41) – 3(24) + 7(-19) = 205 – 72 – 133 = 0\). Confirmed.

Example 4:

If \(A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}\), show that \(|3A| = 27|A|\).

Solution: \(|A| = 1(4-0) = 4\). So \(27|A| = 108\). \(|3A| = \begin{vmatrix} 3 & 0 & 3 \\ 0 & 3 & 6 \\ 0 & 0 & 12 \end{vmatrix} = 3(36-0) = 108\). Hence \(|3A| = 3^3 |A|\).

Example 5:

Prove \(\begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} = (a-b)(b-c)(c-a)\).

Solution: Perform \(R_2 \to R_2 – R_1\) and \(R_3 \to R_3 – R_1\): \(\begin{vmatrix} 1 & a & bc \\ 0 & b-a & c(a-b) \\ 0 & c-a & b(a-c) \end{vmatrix} = (b-a)(c-a) \begin{vmatrix} 1 & a & bc \\ 0 & 1 & -c \\ 0 & 1 & -b \end{vmatrix}\) \(= (b-a)(c-a)[-b – (-c)] = (b-a)(c-a)(c-b) = (a-b)(b-c)(c-a)\).

Here is a detailed explanation with examples for each property of determinants. These examples are designed for \(2 \times 2\) and \(3 \times 3\) matrices where applicable, to help students visualize the concepts clearly.

Concept 2: Properties of Determinants

Properties help in simplifying determinants without direct expansion. These are extremely important for solving problems quickly in CBSE and ISC board examinations.

Key Properties

1. Reflection Property: The value of a determinant remains unchanged if its rows and columns are interchanged: \(|A| = |A^T|\).
   Example (2×2): \(A = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix}, |A| = -1; A^T = \begin{bmatrix} 2 & 5 \\ 3 & 7 \end{bmatrix}, |A^T| = -1\).
   Example (3×3): \(B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}, |B| = 24; B^T = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 4 & 0 \\ 3 & 5 & 6 \end{bmatrix}, |B^T| = 24\).
2. Switching Property: If any two rows (or columns) are interchanged, the sign of the determinant changes.
   Example (2×2): \(C = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, |C| = -2\). Interchange \(R_1,R_2\): \(|C’| = 2 = -|C|\).
   Example (3×3): \(D = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 4 & 0 \end{bmatrix}, |D| = -24\); interchange \(C_1,C_2\): \(|D’| = 24 = -|D|\).
3. Repetition Property: If any two rows (or columns) are identical or proportional, the determinant is zero.
   Example (2×2): \(\begin{bmatrix} a & b \\ a & b \end{bmatrix} = 0\). Example (3×3): \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{bmatrix} = 0\).
4. Scalar Multiple Property: If each element of a row (or column) is multiplied by constant \(k\), determinant multiplies by \(k\).
   Example: \(G = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, |G|=-2\); multiply \(R_1\) by 3: \(\begin{bmatrix} 3 & 6 \\ 3 & 4 \end{bmatrix}, |G’| = -6 = 3|G|\).
5. Common Factor Property: If a common factor exists in any row or column, it can be taken out.
   Example: \(K = \begin{bmatrix} 3 & 6 & 9 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = 3\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\), determinant = 0.
6. Invariance Property (Row/Column Operation): \(R_i \to R_i + kR_j\) or \(C_i \to C_i + kC_j\) leaves determinant unchanged.
   Example: \(L = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, |L|=-2\); \(R_2 \to R_2 – 3R_1\): \(\begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix}, |L’| = -2\).
7. Zero Row/Column Property: All elements of a row/column zero ⇒ determinant zero.
   Example: \(\begin{bmatrix} 0 & 0 \\ 5 & 7 \end{bmatrix} = 0\).
8. Proportional Rows/Columns Property: Two rows/columns proportional ⇒ determinant zero.
   Example: \(Q = \begin{bmatrix} 2 & 4 \\ 3 & 6 \end{bmatrix} = 0\).
9. Factor Multiplication Property: If all elements multiplied by \(k\) for \(n \times n\) matrix, determinant multiplied by \(k^n\).
   Example (2×2): \(\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} = 2^2 \times (-2) = -8\).
10. Sum Property (Linearity): Determinant splits if a row is sum of two terms.
    Example: \(\begin{bmatrix} a+b & c+d \\ e & f \end{bmatrix} = \begin{bmatrix} a & c \\ e & f \end{bmatrix} + \begin{bmatrix} b & d \\ e & f \end{bmatrix}\).
11. Triangular Matrix Property: Upper or lower triangular ⇒ determinant = product of diagonal entries.
    Example: \(\begin{bmatrix} 2 & 1 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} = 48\).
12. Identity Matrix Property: \(|I_n| = 1\).
13. Determinant of Product: \(|AB| = |A||B|\).
14. Determinant of Inverse: \(|A^{-1}| = 1/|A|\).

Solved Examples (Concept 2)

Example 1:

Without expanding, show that \(\begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{vmatrix} = 0\).
Solution: In \(R_1\), take 6 common: \(6 \begin{vmatrix} 17 & 3 & 6 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{vmatrix}\). Since \(R_1 = R_3\), determinant = \(6(0)=0\).

Example 2:

Evaluate \(\begin{vmatrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{vmatrix}\) using properties.
Solution: \(C_3 \to C_3 – 9C_2\): \(\begin{vmatrix} 2 & 7 & 2 \\ 3 & 8 & 3 \\ 5 & 9 & 5 \end{vmatrix}\). Since \(C_1 = C_3\), value is 0.

Example 3:

Using properties, prove \(\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix} = (a+b+c)^3\).
Solution: \(R_1 \to R_1+R_2+R_3\): factor \((a+b+c)\), then row operations lead to lower triangular determinant, total \((a+b+c)^3\).

Example 4:

If \(A\) is a skew-symmetric matrix of order 3, prove \(|A| = 0\).
Solution: \(A^T = -A \Rightarrow |A^T| = |-A| = (-1)^3|A| = -|A|\). Also \(|A^T| = |A| \Rightarrow |A| = -|A| \Rightarrow 2|A|=0\).

Example 5:

Find \(|A|\) if \(A = \begin{bmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{bmatrix}\).
Solution: \(R_1 \to R_1+R_2\): \((x+y+z)\begin{vmatrix} 1 & 1 & 1 \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} = 0\).

Self Exercise & HOTS

1. If \(a, b, c\) are in A.P., find the value of \(\begin{vmatrix} x+2 & x+3 & x+2a \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c \end{vmatrix}\).
2. Prove that \(\begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{vmatrix} = 4xyz\).
3. If \(x, y, z\) are non-zero real numbers, then find the inverse of matrix \(A = \operatorname{diag}(x, y, z)\) using determinants.
4. Show that \(\begin{vmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{vmatrix} = 2abc(a+b+c)^3\).
5. If \(A\) is a square matrix of order 3 and \(|A| = 5\), find \(|\operatorname{adj} A|\).
6. Without expanding, show that \(\begin{vmatrix} \sin^2 A & \cos^2 A & 1 \\ \sin^2 B & \cos^2 B & 1 \\ \sin^2 C & \cos^2 C & 1 \end{vmatrix} = 0\).
7. Solve for \(x\): \(\begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix} = 0\).
8. Let \(A\) be a \(3 \times 3\) matrix such that \(|A| = 2\). Find the value of \(|2 \cdot \operatorname{adj}(2A)|\).
9. If \(a, b, c\) are real numbers and \(\Delta = \begin{vmatrix} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{vmatrix} = 0\), show that either \(a+b+c=0\) or \(a=b=c\).
10. Prove that the value of a determinant of a skew-symmetric matrix of even order is always a perfect square.
11. Prove that: \(\begin{vmatrix} 1 & 1 & 1 \\ \alpha & \beta & \gamma \\ \alpha^3 & \beta^3 & \gamma^3 \end{vmatrix} = (\alpha – \beta)(\beta – \gamma)(\gamma – \alpha)(\alpha + \beta + \gamma)\).
12. If \(\alpha, \beta, \gamma\) are angles of a triangle (\(\alpha + \beta + \gamma = \pi\)), prove: \(\begin{vmatrix} \sin^2 \alpha & \sin \alpha \cos \alpha & \cos^2 \alpha \\ \sin^2 \beta & \sin \beta \cos \beta & \cos^2 \beta \\ \sin^2 \gamma & \sin \gamma \cos \gamma & \cos^2 \gamma \end{vmatrix} = -\sin(\alpha – \beta) \sin(\beta – \gamma) \sin(\gamma – \alpha)\).
13. Show that the following determinant is independent of \(\alpha, \beta, \gamma\) and find its constant value: \(\Delta = \begin{vmatrix} \cos(\alpha+\beta) & -\sin(\alpha+\beta) & \cos 2\beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \end{vmatrix}\).
14. If \(\alpha, \beta, \gamma\) are the roots of \(x^3 + px + q = 0\), evaluate \(\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix}\).
15. Prove: \(\begin{vmatrix} (\beta + \gamma)^2 & \alpha^2 & \alpha^2 \\ \beta^2 & (\gamma + \alpha)^2 & \beta^2 \\ \gamma^2 & \gamma^2 & (\alpha + \beta)^2 \end{vmatrix} = 2\alpha\beta\gamma(\alpha + \beta + \gamma)^3\).
16. Without expanding, prove: \(\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix} = 0\).
17. If \(x, y, z\) are in A.P., evaluate \(\begin{vmatrix} p+2 & p+3 & p+x \\ p+3 & p+4 & p+y \\ p+4 & p+5 & p+z \end{vmatrix}\).
18. Prove: \(\begin{vmatrix} 1 & a & a^2-bc \\ 1 & b & b^2-ca \\ 1 & c & c^2-ab \end{vmatrix} = 0\).
19. Using properties, show: \(\begin{vmatrix} x+k & y & z \\ x & y+k & z \\ x & y & z+k \end{vmatrix} = k^2(x+y+z+k)\).
20. If \(a, b, c\) are all distinct and \(\begin{vmatrix} a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3 \end{vmatrix} = 0\), prove \(abc = -1\).
21. Evaluate: \(\begin{vmatrix} \sin^2 \theta & \cos^2 \theta & 1 \\ \cos^2 \theta & \sin^2 \theta & 1 \\ -10 & 12 & 2 \end{vmatrix}\).
22. Show: \(\begin{vmatrix} y+z & x & x \\ y & z+x & y \\ z & z & x+y \end{vmatrix} = 4xyz\).
23. Prove: \(\begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} = 2(a+b+c)^3\).
24. If \(\Delta = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}\), prove \(\Delta = (a-b)(b-c)(c-a)(a+b+c)\).
25. Without expanding, show \(\begin{vmatrix} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0 \end{vmatrix} = 0\).
26. Find \(k\) if \(\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix} = (a+b+c)^k\).
27. Prove: \(\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc\left(1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)\).
28. Evaluate: \(\begin{vmatrix} \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \\ 1 & \omega & \omega^2 \end{vmatrix}\) where \(\omega\) is a complex cube root of unity.
29. If \(A\) is a square matrix of order 3 and \(|A| = 5\), find \(|3A|\).
30. Prove: \(\begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ac & bc & c^2+1 \end{vmatrix} = 1 + a^2 + b^2 + c^2\).