CBSE Class 12 Mathematics Determinants Notes Set 01

BASIC CONCEPTS

1. Determinant: Every square matrix can be associated to an expression or a number which is known as its determinant.

Determinant of square matrix \( A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \) is given by
\( | A | = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21} \)

and determinant of a matrix \( A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} \) is given by
\( | A | = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1 \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1 \begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1 \begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix} \)

This is known as the expansion of \( | A | \) along first row.
In fact, \( | A | \) can be expanded along any of its rows or columns.

2. Singular and Non-singular Matrix: A square matrix is a singular matrix if its determinant is zero. Otherwise, it is a non-singular matrix.

3. (i) Minor: Let \( A = [a_{ij}] \) be a square matrix of order \( n \). Then the minor \( M_{ij} \) of \( a_{ij} \) in \( A \) is the determinant of the sub-matrix of order \( (n - 1) \) obtained by leaving \( i \text{th} \) row and \( j \text{th} \) column of \( A \).

For example, if \( A = \begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \\ 2 & -4 & 3 \end{bmatrix} \), then
\( M_{11} = \begin{vmatrix} 2 & -1 \\ -4 & 3 \end{vmatrix} = 2, M_{12} = \begin{vmatrix} -3 & -1 \\ 2 & 3 \end{vmatrix} = -7 \) and so on.

(ii) Cofactor: The cofactor \( C_{ij} \) of \( a_{ij} \) in \( A = [a_{ij}]_{n \times n} \) is equal to \( (-1)^{i+j} \) times \( M_{ij} \).

For example, if \( A = \begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \\ 2 & -4 & 3 \end{bmatrix} \), then
\( C_{11} = (-1)^{1+1} M_{11} = M_{11} = 2 \) and \( C_{12} = (-1)^{1+2} M_{12} = -M_{12} = 7 \) and so on

4. Some Important Properties of Determinants:

• (i) Let \( A = [a_{ij}] \) be a square matrix of order \( n \), then the sum of the product of elements of any row (column) with their cofactors is always equal to \( |A| \) or, \( \text{det} (A) \), i.e.,
  \( \sum_{j=1}^{n} a_{ij} C_{ij} = |A| \) and \( \sum_{i=1}^{n} a_{ij} C_{ij} = |A| \)
• (ii) Let \( A = [a_{ij}] \) be a square matrix of order \( n \), then the sum of the product of elements of any row (column) with cofactors of the corresponding elements of some other row (column) is zero, i.e.,
  \( \sum_{j=1}^{n} a_{ij} C_{kj} = 0 \) and \( \sum_{i=1}^{n} a_{ij} C_{ik} = 0, i \neq k \) or \( j \neq k \)
• (iii) Let \( A = [a_{ij}] \) be a square matrix of order \( n \), then \( |A| = |A^T| \).
  In other words, we say that the value of a determinant remains unchanged, if its rows and columns are interchanged.
• (iv) Let \( A = [a_{ij}] \) be a square matrix of order \( n (\geq 2) \) and \( B \) be a matrix obtained from \( A \) by interchanging any two rows (columns) of \( A \), then \( |B| = -|A| \).
• (v) If any two rows (columns) of a square matrix \( A = [a_{ij}] \) of order \( n (\geq 2) \) are identical, then value of its determinant is zero i.e., \( |A| = 0 \).
• (vi) Let \( A = [a_{ij}] \) be a square matrix of order \( n \), and let \( B \) be the matrix obtained from \( A \) by multiplying each element of a row (column) of \( A \) by a scalar \( k \), then \( |B| = k|A| \).
• (vii) Let \( A \) be a square matrix such that each element of a row (column) of \( A \) is expressed as the sum of two or more terms. Then the determinant of \( A \) can be expressed as the sum of the determinants of two or more matrices of the same order.
• (viii) Let \( A \) be a square matrix and \( B \) be a matrix obtained from \( A \) by adding to a row (column) of \( A \) a scalar multiple of another row (column) of \( A \), then \( |B| = |A| \).
• (ix) Let \( A \) be a square matrix of order \( n (\geq 2) \) such that each element in a row (column) of \( A \) is zero, then \( |A| = 0 \).
• (x) If \( A = [a_{ij}] \) is a diagonal matrix of order \( n (\geq 2) \), then
  \( |A| = a_{11} \cdot a_{22} \cdot a_{33} \dots a_{nn} \) i.e., \( |A| \) is the product of its diagonal elements.
• (xi) If \( A \) and \( B \) are square matrices of the same order, then
  \( |AB| = |A| |B| \)
• (xii) If \( A = [a_{ij}] \) is a triangular matrix of order \( n \), then
  \( |A| = a_{11} \cdot a_{22} \cdot a_{33} \dots a_{nn} \) i.e., \( |A| \) is the product of its diagonal elements.
• (xiii) If \( A = [a_{ij}] \) is a square matrix of order \( n \), then \( |kA| = k^n|A| \), because \( k \) is common from each row (or column) of \( kA \).
• (xiv) We can take out any common factor from any one row or any one column of a given determinant.

5. Area of a triangle with vertices \( (x_1, y_1), (x_2, y_2) \) and \( (x_3, y_3) \) is given by
\( \Delta = \text{Numerical value of } \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)
Note: Since area is positive quantity therefore we take absolute value of \( \Delta \).

6. (i) If \( A \) is a skew-symmetric matrix of odd order, then \( |A| = 0 \).
(ii) The determinant of a skew-symmetric matrix of even order is a perfect square.

7. Some Important Facts:

• (i) Only square matrices have determinants.
• (ii) We cannot equate the corresponding elements of equal determinants like matrices
  i.e., \( \begin{vmatrix} x & y \\ z & w \end{vmatrix} = \begin{vmatrix} l & m \\ n & p \end{vmatrix} \nRightarrow \begin{matrix} x = l, & y = m \\ z = n, & w = p \end{matrix} \)
• (iii) In the case of matrices. We take out any common factor from each elements of matrix, while in the case of determinants we can take out common factor from any one row or any one column of the determinant.
• (iv) If the value of determinant '\( \Delta \)' becomes zero by substituting \( x = a \) then \( (x - a) \) is factor of the determinant '\( \Delta \)'.
• (v) If area is given then both positive and negative values of the determinant is taken for calculation.
• (vi) To prove three points collinear, we show area of the triangle formed by these three points is zero.