CBSE CLASS 12 MATHEMATICS 041, CH 04 DETERMINANTS PREVIOUS YEAR QUESTIONS

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determinant is a special scalar (numerical) value calculated from the elements of a square matrix. It provides crucial insights into the matrix, acting as a scaling factor for linear transformations, and determining whether a system of linear equations has a unique solution. [1, 2, 3]

Key Concepts & Properties

Applicability: Determinants can only be calculated for square matrices (e.g., \(2 \times 2\), \(3 \times 3\)).

Notation: Denoted as \(\det(A)\), \(\det A\), or \(|A|\).

Invertibility: If \(\det(A) \neq 0\), the matrix is invertible (nonsingular). If \(\det(A) = 0\), the matrix is singular and cannot be inverted. [1, 2, 3, 4, 5]

Calculating a \(2 \times 2\) Matrix

For a generic \(2 \times 2\) matrix:
\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)

The determinant is calculated as:
\(\det(A) = (ad) - (bc)\)

Calculating a \(3 \times 3\) Matrix

For a generic \(3 \times 3\) matrix:
\(A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\)

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