CBSE Class 12 Mathematics Determinants Notes Set 03

Important Notes and Formulas

Definition

The symbol \( \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \) is called the determinant of order two. Its value is given by:
\( D = a_1 b_2 - a_2 b_1 \)

Expansion of Determinant

The symbol \( \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \) is called the determinant of order three. Its value is
\( (a_1 b_2 c_3 + b_1 c_2 a_3 + c_1 a_2 b_3) - (a_3 b_2 c_1 + b_3 c_2 a_1 + c_3 a_2 b_1) \)
Its value can be found as:
\( D = a_1 \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1 \begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1 \begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix} \)
(Expanded by R₁)
\( D = a_1 \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & c_1 \\ b_3 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix} \)
(Expanded by C₁)
In this manner we can expand a determinant in 6 ways using the elements of R₁, R₂, R₃ or C₁, C₂, C₃.

Application of Determinants

(i) Area of the triangle whose vertices are \( (x_r, y_r) \); r = 1, 2, 3, is:
\( D = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \) (the absolute value of D is taken as the area)
If D = 0, then the three points are collinear.
(ii) The equation of the straight line passing through \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( \begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0 \)

Minors

The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row and the column in which the given element stands. For example,
the minor of \( a_1 \) in \( \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \) is \( \begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} \) and the minor of \( b_2 \) is \( \begin{vmatrix} a_1 & c_1 \\ a_3 & c_3 \end{vmatrix} \).
Hence a determinant of order two will have 4 minors and a determinant of order three will have 9 minors.
We denote the minor of the (i, j)^th element by \( M_{ij} \).

Cofactor

The cofactor of the element \( a_{ij} \) is \( C_{ij} = (-1)^{i+j} M_{ij} \), where i and j denote the row and column in which the particular element lies and \( M_{ij} \) is the minor of the (i, j)^th element.
Note that the value of a determinant of order three in terms of minors and cofactors can be written as:
\( D = a_{11} M_{11} - a_{12} M_{12} + a_{13} M_{13} \)   or
\( D = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13} \) and so on.

Adjoint

The transpose of the cofactor matrix is called the adjoint matrix.

Properties of Adjoint Matrix

(i) \( A \cdot (\text{adj } A) = |A| I_n = (\text{adj } A) \cdot A \)
(ii) \( |\text{adj } A| = |A|^{n-1} \), where n is the order of the square matrix A

Inverse of a Matrix

\( A^{-1} = \frac{1}{|A|} \text{adj}(A) \), provided |A| ≠ 0

Properties of Inverse Matrices

(i) \( (A^T)^{-1} = (A^{-1})^T \)
(ii) \( (AB)^{-1} = B^{-1} A^{-1} \)
(iii) \( (A^{-1})^{-1} = A \)
(iv) \( (KA)^{-1} = \frac{1}{K} A^{-1} \), if K ≠ 0
(v) Let A, B, C be square matrices of the same order n. If A is a non-singular matrix, then
    (a) AB = AC ⇒ B = C (left cancellation law)
    (b) BA = CA ⇒ B = C (right cancellation law)
(vi) If A is a non-singular matrix such that A is symmetric, then \( A^{-1} \) is also symmetric.

Consistent and Inconsistent Systems

(1) Consistent: Solution exists, whether unique or infinite in number.
(2) Inconsistent: Solution does not exist.

Solution of a System of Linear Equations by Matrix Method

Consider a system of linear equations
\( a_{11} x_1 + a_{12} x_2 + \ldots + a_{1n} x_n = b_1 \)
\( a_{21} x_1 + a_{22} x_2 + \ldots + a_{2n} x_n = b_2 \)
……………………………………
\( a_{n1} x_1 + a_{n2} x_2 + \ldots + a_{nn} x_n = b_n \)
We can express these equations as a single matrix equation
\( \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} \),  i.e. AX = B
Let |A| ≠ 0, so that \( A^{-1} \) exists uniquely. Pre-multiplying both sides of AX = B by \( A^{-1} \), we get
\( A^{-1}(AX) = A^{-1}B \) ⇒ \( (A^{-1}A)X = A^{-1}B \)
⇒ \( IX = A^{-1}B \) ⇒ \( X = A^{-1}B \)

Criterion of Consistency

Let AX = B be a system of n linear equations in n variables.
(i) If |A| ≠ 0, then the system of equations is consistent and has a unique solution given by \( X = A^{-1}B \).
(ii) If |A| = 0 and (adj A) B = O, then the system of equations is consistent and has infinitely many solutions, except the case of parallel lines when there is no solution.
(iii) If |A| = 0 and (adj A) B ≠ O, then the system of equations is inconsistent, i.e. it has no solution.