Determinants of Matrices

• Dr. Gabriel Obed Fosu from the Department of Mathematics at Kwame Nkrumah University of Science and Technology presents notes on Determinants

Determinants of Matrices

• Determinants are scalar values of square matrices, characterizing some properties.
• The determinant of matrix A can be denoted as det(A), det A, or |A|.
• Determinants are used to characterize nonsingular matrices, express solutions of nonsingular systems (Ax = b), and express vector cross products.

Calculating Determinants

• Mij represents the determinant of the submatrix formed by removing the i-th row and j-th column from matrix A.
• The determinant of an n×n matrix A can be calculated using the first-row Laplace expansion: |A| = ∑((-1)^(1+j) * a1j * M1j) from j=1 to n.
• Mᵢⱼ values are also known as minors.
• The determinant of a 2 × 2 matrix A = [[a11 a12], [a21 a22]] is |A| = a11a22 - a12a21, the minors being a22 and a21
• The determinant of a 3 x 3 Matrix is calculated as follows: |A| = a11 * (a22 * a33 − a23 * a32) − a12 * (a21 * a33 − a23 * a31) + a13 * (a21 * a32 − a22 * a31)
• Determinants can be expanded along any row or column, not just the first row.
• The i-th row expansion is |A| = ∑((-1)^(i+j) * aij * Mij) from j=1 to n.
• The j-th column expansion is |A| = ∑((-1)^(i+j) * aij * Mij) from i=1 to n.
• The expression (−1)^(i+j) follows a chessboard pattern of signs.

Cofactors, Adjoint, and Inverse Matrices

• The (i, j) cofactor of A, denoted by Cij, is defined as Cij = (−1)^(i+j) * Mij. The adjoint of A (adj A) is the transpose of the matrix of cofactors.
• The inverse of a matrix is given by A⁻¹ = (1/det A) * adj A.

Properties of Determinants

• The product of a matrix and its adjoint is a diagonal matrix with diagonal entries equal to det(A); A * adj(A) = det(A) * I
• The determinant of a lower triangular matrix is the product of its diagonal elements, also applicable to upper triangular and diagonal matrices
• A matrix and its transpose have equal determinants: |A| = |AT|

Rules of Determinants

• If a row or column of a matrix is all zeros, the determinant is 0.
• For an n×n matrix A and a scalar c; det(cA) = cⁿ * det(A).
• The determinants of AB is det(AB) = (det A) * (det B)
• A determinant is a linear function of each row separately; if two rows are added the determinants are added as well.
• If multiplying a row of A by a scalar t, then the determinant of the modified matrix is t*det A.
• The determinant changes sign if two rows of a matrix are exchanged.
• If a multiple of a row is subtracted from another row, the determinant remains unchanged.
• When two rows of a matrix equal, the determinant is zero

Theorems

• A matrix A is nonsingular if and only if det A ≠ 0.
• A matrix A is singular if and only if det A = 0.
• The homogeneous system Ax = 0 has a nontrivial solution if and only if det A = 0

Cramer's Rule

• Cramer's Rule applies to an n×n invertible matrix A.
• Let b be a column vector with n components, and let Aᵢ be A with the i-th column replaced by b.
• If x is the solution to Ax = b, then xi = det(Ai) / det(A) for i = 1, 2, ..., n.

Alternative Determinant Method for 3x3 Matrix

• Copy the first two columns of the matrix to the right of the matrix.
• Calculate the products of the elements along the six diagonals.
• Assign plus signs to downward-sloping diagonals and minus signs to upward-sloping diagonals.
• The determinant is the sum of the products with plus signs minus the sum of the products with minus signs.

Lecture notes on determinants of matrices, including calculation methods and properties. Determinants are scalar values of square matrices used to characterize properties. The determinant of an n×n matrix A can be calculated using the first-row Laplace expansion.