Determinant of a matrix

Learning Objectives Define the determinant of a square matrix. Calculate the determinant of a 2x2 matrix using the standard formula. Calculate the determinant of a 3x3 matrix using the diagonal method (Sarrus's Rule). Calculate the determinant of a 3x3 matrix using cofactor expansion. Define and identify minors and cofactors of a matrix element. Distinguish between singular and non-singular matrices based on their determinant's value. How can a single number tell you the area of a transformed shape or if a system of equations has a unique solution? 🤔 Let's find out! The determinant is a special scalar value that can be calculated from a square matrix. It provides crucial information about the matrix, such as whether it is invertible, and has significant appl...

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 = [[2, 7], [1, 4]] is a 2x2 square matrix. DeterminantA unique scalar value associated with a square matrix. It is denoted by det(A) or |A|.For matrix A = [[2, 1], [3, 4]], the determinant is det(A) = (2)(4) - (1)(3) = 5. Minor (M_ij)The determinant of the submatrix formed by deleting the i-th row and j-th column of the original matrix.For A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], the minor M_11 is the determinant of the submatrix [[5, 6], [8, 9]], which is (5)(9) - (6)(8) = -3. Cofactor (C_ij)A 'signed' minor, calculated using the formula C_ij = (-1)^(i+j) * M_ij. The sign depends on the position of the element.Using the...

Determinant of a 2x2 Matrix If A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, then det(A) = |A| = ad - bc To find the determinant of a 2x2 matrix, 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 of a 3x3 Matrix (Sarrus's Rule) If A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}, then det(A) = (aei + bfg + cdh) - (gec + hfa + idb) This is a mnemonic for 3x3 matrices only. Augment the matrix with its first two columns. Sum the products of the three main diagonals, then subtract the sum of the products of the three anti-diagonals. Determinant by Cofactor Expansion For an n x n matrix A, th...