Revision: Applications of Matrices and Determinants Business Mathematics and Statistics HSC Commerce Class 12 Tamil Nadu Board of Secondary Education

A determinant is a scalar value associated with a square matrix.

For a \[ 2 \times 2 \] matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

its determinant is written as: \[ |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

and calculated as: \[ |A| = ad - bc \]

Remember: A determinant is a number, while a matrix is an arrangement of numbers.

Cramer’s Rule

For the system: \[ a_1x + b_1y = c_1 \], \[ a_2x + b_2y = c_2 \]

\[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \]

\[ D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} \]

\[ D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \]

If \[ D \neq 0, \] then the system has a unique solution:

\[ x = \frac{D_x}{D} \]

\[ y = \frac{D_y}{D} \]

• A determinant is a number associated with a square matrix.
• For \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix}, \] the value is \[ ad - bc. \]
• Cramer’s Rule solves a pair of linear equations using determinants.
• \[ D \] is formed from the coefficients of \[ x \] and \[ y. \]
• \[ D_x \] replaces the \[ x\text{-coefficient} \] column with constants.
• \[ D_y \] replaces the \[ y\text{-coefficient} \] column with constants.
• If \[ D \neq 0, \] the solution is \[ x = \frac{D_x}{D}, y = \frac{D_y}{D}. \]
• If \[ D = 0, \] Cramer’s Rule does not give a unique solution.