Determinants PDF

# Determinents ## Introduction - A determinant is a numerical value that can be calculated from a square matrix. - It is related to the matrix. ## Calculation of Determinant - A determinant of a 1x1 matrix is simply the value a11. - A determinant of a 2x2 matrix is calculated using the followi...

# Determinents ## Introduction - A determinant is a numerical value that can be calculated from a square matrix. - It is related to the matrix. ## Calculation of Determinant - A determinant of a 1x1 matrix is simply the value a11. - A determinant of a 2x2 matrix is calculated using the following formula: $ |A| = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}$ - A determinant of a 3x3 matrix is calculated using the following formula: $ |A| = \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix} $ ## Properties of Determinents - The determinant of a matrix is equal to the determinant of its transpose. - If a matrix has two rows or columns identical, the determinant is zero. - If a row (or column) of a determinant is multiplied by a scalar, the determinant is multiplied by the scalar. - If two rows (or columns) of a determinant are interchanged, the determinant changes its sign. - If a multiple of one row (or column) of a determinant is added to another row (or column) of it, the value of the determinant remains the same. - The determinant of a triangular matrix is equal to the product of its diagonal elements. ## Applications of Determinents - Determinents are used to solve linear equations. - They can also be used to find the inverse of a matrix. - Determinents are used in finding the eigenvalues of a matrix. - They can also be used to find the area of a triangle or parallelogram. ## Difference Between Matrix and Determinant - A **matrix** is a rectangular array of numbers, while a **determinant** is a single number that is calculated from a square matrix. - A **matrix** can have any number of rows and columns, while a **determinant** can only be calculated for a square matrix. ## Example The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by the following determinant: $ \triangle = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} $