Determinants Class 12 Chapter Notes

Key Definitions

Determinant

A scalar value associated with every square matrix. It is denoted by det(A) or |A|.

Minor (Mij)

The determinant of the square submatrix obtained by deleting the i-th row and j-th column of the original matrix.

Cofactor (Aij or Cij)

Defined as $(-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor of the element $a_{ij}$.

Singular Matrix

A square matrix A is called singular if its determinant is zero, i.e., $|A| = 0$. A singular matrix does not have an inverse.

Non-Singular Matrix

A square matrix A is called non-singular if its determinant is non-zero, i.e., $|A| \neq 0$. A non-singular matrix always has an inverse.

Adjoint of a Matrix (adj A)

The transpose of the cofactor matrix of A. If $C = [A_{ij}]$ is the cofactor matrix, then adj A = $C^T$.

Inverse of a Matrix ($A^{-1}$)

For a non-singular square matrix A, its inverse is given by $A^{-1} = \frac{1}{|A|} \text{adj A}$. The inverse exists only if $|A| \neq 0$.

Key Points & Formulas

• Determinant of a 1x1 matrix: For A = $[a]$, $|A| = a$.
• Determinant of a 2x2 matrix: For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $|A| = ad - bc$.
• Expansion of a 3x3 matrix: Can be expanded along any row or column using cofactors. $|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$ (along R1).
• Area of a Triangle: If vertices are $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, Area = $1/2 \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$. Area is always positive; if result is negative, take its absolute value.
• Condition for collinear points: Three points are collinear if the area of the triangle formed by them is zero.
• Property: For any square matrix A of order n, $A(\text{adj A}) = (\text{adj A})A = |A|I$, where I is the identity matrix of order n.
• Property: $|\text{adj A}| = |A|^{n-1}$.
• Property: $|AB| = |A||B|$ for square matrices A and B of the same order.
• System of Linear Equations (Matrix Method): For $AX=B$, if $|A| \neq 0$, then $X = A^{-1}B$ (unique solution). If $|A| = 0$ and $( ext{adj A})B \neq O$, then no solution (inconsistent). If $|A| = 0$ and $( ext{adj A})B = O$, then infinitely many solutions or no solution (consistent/inconsistent, further investigation needed).

Properties of Determinants (for Exam Success)

Understanding and applying the properties of determinants is crucial for solving complex problems efficiently, especially in board exams where proof-based questions are common. These properties allow simplification of determinants without direct expansion.

1. Row/Column Interchange: If any two rows (or columns) of a determinant are interchanged, the sign of the determinant changes. For example, if $R_i \leftrightarrow R_j$, then $|A'| = -|A|$.
2. Identical Rows/Columns: If any two rows (or columns) of a determinant are identical (or proportional), then the value of the determinant is zero. This is a very common trick used in proofs.
3. Scalar Multiplication: If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the determinant gets multiplied by k. This means $k \begin{vmatrix} a & b \\ c & d \end{vmatrix} \neq \begin{vmatrix} ka & kb \\ kc & kd \end{vmatrix}$, but $k \begin{vmatrix} a & b \\ c & d \end{vmatrix} = \begin{vmatrix} ka & kb \\ c & d \end{vmatrix}$. Also, for a matrix A of order n, $|kA| = k^n |A|$.
4. Sum of Terms: If elements of a row (or a column) are expressed as a sum of two or more terms, the determinant can be expressed as the sum of two or more determinants. For example, $\begin{vmatrix} a+x & b \\ c+y & d \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} x & b \\ y & d \end{vmatrix}$.
5. Row/Column Operations: If to any row (or column) of a determinant, the elements of any other row (or column) are added after multiplying by a scalar quantity k, the value of the determinant remains unchanged. This operation is denoted as $R_i \to R_i + kR_j$ or $C_i \to C_i + kC_j$. This property is extensively used to make elements zero for easier expansion.
6. Transpose Invariance: The value of a determinant remains unchanged if its rows and columns are interchanged, i.e., $|A^T| = |A|$. This implies that any property true for rows is also true for columns.

Applying these properties judiciously can transform a complicated determinant into a much simpler form, often resulting in a determinant with many zeros, making expansion straightforward.

Steps to Find the Inverse of a Matrix by Adjoint Method

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Worked Examples

• Example 1: Calculate the determinant of a 2x2 matrix. If $A = \begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}$, then $|A| = (5 \times 4) - (2 \times 3) = 20 - 6 = 14$.
• Example 2: Find the minor and cofactor for an element in a 3x3 matrix. For $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$, let's find the minor and cofactor of $a_{12}=2$. Minor $M_{12} = \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = (4 \times 9) - (6 \times 7) = 36 - 42 = -6$. Cofactor $A_{12} = (-1)^{1+2} M_{12} = (-1)^3 (-6) = (-1)(-6) = 6$.
• Example 3: Find the inverse of a 2x2 matrix. Given $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$. 1. $|A| = (2 \times 4) - (1 \times 3) = 8 - 3 = 5$. Since $|A| \neq 0$, $A^{-1}$ exists. 2. Cofactor Matrix: $A_{11}=4, A_{12}=-3, A_{21}=-1, A_{22}=2$. So, $C = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$. 3. adj A = $C^T = \begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix}$. 4. $A^{-1} = \frac{1}{|A|} \text{adj A} = \frac{1}{5} \begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} 4/5 & -1/5 \\ -3/5 & 2/5 \end{bmatrix}$.

YoLearn Exam Tip

Sign Errors with Cofactors: A very common mistake in determinant problems, especially for 3x3 matrices, is miscalculating the sign of cofactors. Remember the pattern for $(-1)^{i+j}$: it alternates like a checkerboard (+ - + / - + - / + - +). Always double-check your signs before proceeding. For properties, remember that operations like $R_i \to R_i + kR_j$ do NOT change the determinant's value, but $R_i \leftrightarrow R_j$ does change the sign, and multiplying a row by k multiplies the determinant by k. Differentiating these is key to avoiding errors in derivations and proofs.

Practice Questions with Solutions

• Q: What is the condition for a square matrix A to be invertible? A: A square matrix A is invertible if and only if it is non-singular, i.e., its determinant $|A|$ is not equal to zero ($|A| \neq 0$).
• Q: How does interchanging two adjacent rows of a determinant affect its value? A: Interchanging any two rows (or columns) of a determinant changes its sign. So, if the original determinant was D, the new determinant will be -D.
• Q: If all elements of a row of a determinant are multiplied by a constant 'k', how is the value of the determinant affected? A: The value of the determinant gets multiplied by 'k'. For example, if one row is multiplied by 'k', the new determinant is k times the original determinant.
• Q: When are three points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ considered collinear using determinants? A: Three points are collinear if the area of the triangle formed by them is zero. This means $1/2 \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0$.

Frequently Asked Questions