Sec 3.6 Determinants 2x2 matrix Evaluate the determinant of

Cramer’s Rule (solve linear system) Sec 3.6 Determinants Cramer’s Rule (solve linear system) Solve the system

Cramer’s Rule (solve linear 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 3x3 matrix signs Find det A

Sec 3.6 Determinants The cofactor expansion of det A along the first row of A Note: 3x3 determinant expressed in terms of three 2x2 determinants 4x4 determinant expressed in terms of four 3x3 determinants 5x5 determinant expressed in terms of five 4x4 determinants nxn determinant expressed in terms of n determinants of size (n-1)x(n-1)

nxn matrix Sec 3.6 Determinants We multiply each element by its cofactor ( in the first row) Also we can choose any row or column Th1: 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 A1 + i-th row A2 Prop 5: i-th col B = i-th col A1 + i-th col A2

Row and Column Properties Either upper or lower Zeros below main diagonal Zeros above main diagonal Prop 6: 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 detA = 0

Determinant and inevitability THM 2: det ( A B ) = det A * det B Note: 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 Def: Adjoint matrix of A Let A =[aij] be an nxn matrix . The cofactor matrix = [Aij] signs Find the cofactor matrix Find the adjoint matrix Def: Adjoint matrix of A

Another method to find the inverse Thm2: The inverse of A Find the inverse of A

Computational Efficiency The amount of labor required to compute a numerical calculation is measured by the number of arithmetical operations it involves Goal: let us count just the number of multiplications required to evaluate an nxn determinant using cofactor expansion 2x2: 2 multiplications 3x3: three 2x2 determinants  3x2= 6 multiplications 4x4: four 3x3 determinants  4x3x2= 24 multiplications 5x5: four 3x3 determinants  4x3x2= 24 multiplications - - - - - - - - - - - - - - - - - - - - - - - - - - - - nxn: n (n-1)x(n-1) determinants  nx…x3x2= n! multiplications

Computational Efficiency Goal: let us count just the number of multiplications required to evaluate an nxn determinant using cofactor expansion nxn: determinants  requires n! multiplications a typical 1998 desktop computer , using MATLAB and performing aonly 40 million operations per second To evaluate a determinant of a 15x15 matrix using cofactor expansion  requires a supercomputer capable of a billion operations per seconds To evaluate a detrminant of a 25x25 matrix using cofactor expansion  requires

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