Matrix Operations: Determinant Determinants • Determinants are only applicable for square matrices. • Determinant of the square matrix 퐴 is denoted as: det(퐴) or 퐴 • Recall that the absolute value of the determinant of a 2 × 2 matrix is equal to the area of parallelogram of the rows of that matrix. • Similarly, the absolute value of the determinant of a 3 × 3 matrix is equal to the volume of parallelepiped of the rows of that matrix. • Therefore, the absolute value of the determinant of a 푛 × 푛 matrix is equal to the n-dimensional volume, constructed by the rows of that matrix. Determinant of a 2 × 2 matrix • Recall that: 풂ퟏ 푎11 푎12 푎11 푎12 퐴 = , 퐴 = = 푎11푎22 − 푎12푎21. 풂ퟐ 푎21 푎22 푎21 푎22 풂2 풂1 Determinant of a 3 × 3 matrix • Also recall the determinant for a 3 ×3 matrix: 푟 푟 푟 풓1 11 12 13 푟 푟 푟 • 푅 = 풓2 21 22 23 풓3 푟31 푟32 푟33 • If the row vectors are linearly dependent, then the determinant is zero, and the matrix is NOT invertible. • Notice if the row vectors are linearly dependent the volume will be zero, as the vectors lie on a plane on a line. Determinant of a 3 × 3 matrix • To compute the determinant of a 3 × 3 matrix,. • The first element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (first) column corresponding to that element from the matrix. • The negate of second element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (second) column corresponding to that element from the matrix. • The third element in the top row is multiplied with the determinant of the sub- matrix resulting from removing the (first) row and the (third) column corresponding to that element from the matrix. 푟11 푟12 푟13 푟22 푟23 푟21 푟23 푟21 푟22 • 푅 = 푟21 푟22 푟23 = 푟11 − 푟12 + 푟13 = 푟32 푟33 푟31 푟33 푟31 푟32 푟31 푟32 푟33 푟11 푟22푟33 − 푟23푟32 − 푟12 푟21푟33 − 푟23푟31 + 푟13 푟21푟32 − 푟22푟31 Determinant of a 3 × 3 matrix / Cofactor • In the determinant of a 3 × 3 matrix, we multiplied the first row elements in their corresponding cofactors. • The cofactor of the element 푖, 푗 of 푛 × 푛 matrix 퐴 is: 푖+푗 퐶푖푗 = (−1) det 푀푖푗 • Where 푀푖푗 is submatrix after removing row 푖 and column 푗. • Determinant of 퐴 is: det 퐴 = 푎푖1퐶푖1 + 푎푖2퐶푖2 + ⋯ + 푎푖푛퐶푖푛 • In the above formula the row 푖 could be any row of 퐴 and it is not necessarily the first row. • In fact it need not be a row. It can be any column j. • (So in order to compute the determinant, it is always wise to choose the row or a column that has most number of zeroes and compute the cofactor of only its non-zero elements.) Determinant properties • The determinant of identity matrix is 1. 퐼 = 1 • The determinant changes sign when two rows are exchanged. 푐 푑 푎 푏 = − 푎 푏 푐 푑 • The determinant is a linear function of each row separately. 푡푎 푡푏 푎 푏 = 푡 푐 푑 푐 푑 푎 + 푎′ 푏 + 푏′ 푎 푏 푎′ 푏′ = + 푐 푑 푐 푑 푐 푑 Determinant properties • If one row is a scalar multiple of another row then det(퐴) = 0 푎 푏 푐 푎 푏 = 0 푑 푒 푓 = 0 푡푎 푡푏 푡푎 푡푏 푡푐 푎 푏 푐 푎 푏 푐 푑 푒 푓 = 0, 푑 푒 푓 = 0 푎 + 푑 푏 + 푒 푐 + 푓 2푎 + 푑 2푏 + 푒 2푐 + 푓 푎 푏 푐 푑 푒 푓 = 0 2푎 + 5푑 2푏 + 5푒 2푐 + 5푓 Determinant properties • Row reduction does not change the determinant of 퐴 푎 푏 푎 푏 = 푐 − 훾푎 푑 − 훾푏 푐 푑 훾 is a non-zero scalar • A matrix with a row of zeros has det(퐴) = 0 푎 푏 = 0 0 0 Determinant properties • If 퐴 is a triangular then the determinant is the product of diagonal elements. 푎 푏 푎 0 = 푎푑, = 푎푑 0 푑 푐 푑 This is also applicable for diagonal matrices: 푎 0 0 0 푏 0 = 푎푏푐 0 0 푐 • If 퐴 is singular (columns or rows are linearly dependent) det(퐴) = 0 • 퐴퐵 = 퐴 퐵 • 퐴푇 = 퐴 Rank of Matrix • Let 푚 = min 푟표푤, 푐표푙푢푚푛 • Rank of matrix is the size of the largest square sub-matrix with non- zero determinant. • Matrix is full-ranked, if its rank = m. • Matrix is rank-deficient, if its rank < m. • It is not possible to have matrix’s rank > m. Sub-Matrix • In order to find the rank of matrix we should find the largest quare sub-matrix with non-zero determinant. • For making a sub-matrix we are allowed to remove rows or columns of a matrix • Example: A is a 5 × 3 matrix • Removing two rows of A 푟표푤1 푟표푤2 푟표푤2 푟표푤3 = 푟표푤4 푟표푤4 푟표푤5 푟표푤5 Matrix Rank • Example: Find the rank of matrix A 0 1 2 퐴 = 1 2 1 2 7 8 Row 1 and Row 2 of matrix A are linearly independent. However Row 3 is a linear combination of Row 1 and 2. 푟표푤3 = 3 × 푟표푤1 + 2 × 푟표푤2 So A only have two independent row vectors. Now let remove third row and first column of A then we have a 2 × 2 matrix which determinant is not zero. 1 2 ≠ 0 2 1 So rank of A is 2. .