Determinant of a matrix

Learning Objectives Define the determinant of a square matrix. Accurately calculate the determinant of a 2x2 matrix. Calculate the determinant of a 3x3 matrix using the method of cofactor expansion. Calculate the determinant of a 3x3 matrix using Sarrus' Rule. Define and calculate the minor and cofactor of an element in a matrix. Use the determinant to identify whether a matrix is singular or non-singular (invertible). Apply the determinant to find the area of a triangle given its vertices. How can a single number unlock secrets about a matrix, like whether a system of equations has a unique solution or how a geometric shape is scaled? 🤔 The determinant is a special scalar value that can be calculated from any square matrix. This powerful number provides crucial inf...

TermDefinitionExample Square MatrixA matrix with an equal number of rows and columns (an n x n matrix). Determinants can only be calculated for square matrices.A = \begin{bmatrix} 2 & 9 \\ 1 & 5 \end{bmatrix} is a 2x2 square matrix. DeterminantA unique scalar value associated with a square matrix, denoted as det(A) or |A|. It provides important information about the matrix, such as its invertibility.For A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}, det(A) = (2)(4) - (1)(3) = 5. Minor (M_ij)The determinant of the submatrix that remains after deleting the i-th row and j-th column of a matrix.For A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, the minor M₁₂ is the determinant of the matrix formed by removing row 1 and column...

Determinant of a 2x2 Matrix For a matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, the determinant is det(A) = ad - bc. This is the fundamental formula for all 2x2 matrices. Multiply the elements of the main diagonal (top-left to bottom-right) and subtract the product of the elements of the other diagonal (top-right to bottom-left). Determinant by Cofactor Expansion For an n x n matrix A, the determinant can be found by expanding along any row i: det(A) = \sum_{j=1}^{n} a_{ij}C_{ij} = a_{i1}C_{i1} + a_{i2}C_{i2} + ... + a_{in}C_{in} This general method works for any size square matrix. Choose a row or column (usually one with zeros to simplify), then multiply each element in that row/column by its corresponding cofactor and sum the results. Sarrus' R...