Sec 3 6 Determinants 2 x 2 matrix

Sec 3. 6 Determinants 2 x 2 matrix Evaluate the determinant of

Sec 3. 6 Determinants Cramer’s Rule (solve linear system) Solve the system

Sec 3. 6 Determinants Cramer’s Rule (solve linear system) Solve the system

Sec 3. 6 Determinants Def: Minors Let A =[aij] be an nxn matrix. The ijth minor of A ( or the minor of aij) is the determinant Mij of the (n-1)x(n-1) submatrix after you delete the ith row and the jth column of A. Find

Sec 3. 6 Determinants Def: Cofactors Let A =[aij] be an nxn matrix. The ijth cofactor of A ( or the cofactor of aij) is defined to be Find signs

Sec 3. 6 Determinants 3 x 3 matrix signs Find det A

Sec 3. 6 Determinants The cofactor expansion of det A along the first row of A Note: q 3 x 3 determinant q 4 x 4 determinant q 5 x 5 determinant q nxn determinant expressed in terms of three 2 x 2 determinants four 3 x 3 determinants five 4 x 4 determinants n determinants of size (n-1)x(n-1)

Sec 3. 6 Determinants nxn matrix We multiply each element by its cofactor ( in the first row) Also we can choose any row or column Th 1: the det of an nxn matrix can be obtained by expansion along any row or column. i-th row j-th row

Row and Column Properties Prop 1: interchanging two rows (or columns)

Row and Column Properties Prop 2: two rows (or columns) are identical

Row and Column Properties Prop 3: (k) i-th row + j-th row (k) i-th col + j-th col

Row and Column Properties Prop 4: (k) i-th row (k) i-th col

Row and Column Properties Prop 5: i-th row B = i-th row A 1 + i-th row A 2 Prop 5: i-th col B = i-th col A 1 + i-th col A 2

Row and Column Properties Either upper or lower Zeros below main diagonal Prop 6: Zeros above main diagonal det( triangular ) = product of diagonal

Row and Column Properties

Transpose Prop 6: det( matrix ) = det( transpose)

Transpose

Determinant and invertibility THM 2: The nxn matrix A is invertible det. A = 0

Determinant and inevitability THM 2: Note: det ( A B ) = det A * det B Proof: Example: compute

Cramer’s Rule (solve linear system) Solve the system

Cramer’s Rule (solve linear system) Use cramer’s rule to solve the system

Adjoint matrix Def: Cofactor matrix Let A =[aij] be an nxn matrix. The cofactor matrix = [Aij] Find the cofactor matrix Def: Adjoint matrix of A signs Find the adjoint matrix

Another method to find the inverse Thm 2: The inverse of A Find the inverse of A