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Instructions for assignment 7 in math 324, which includes evaluating determinants using cofactor expansion and elementary row operations, solving systems of equations using row reduction, inverse matrices, and cramer's rule, and proving properties of determinants for nilpotent, lower triangular, similar, and orthogonal matrices.
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Math 324: Assignment 7
Instructions. The following six exercises plus a written solution to the exercise you select to present in class are due on Monday, November 1.
(^) by:
(a) using the cofactor expansion along the first column;
(b) using the cofactor expansion along the first row.
by
(a) using elementary row operations to make the first entries in rows 2 and 4 both 0, and then using the cofactor expansion along the first column.
(b) using elementary row operations to make the matrix upper triangular.
3 x + y + z = 4 − 2 x − y = 12 x + 2 y + z = − 8
Using MathCAD, solve
this system by
(a) row reducing the agmented matrix;
(b) using the inverse of the coefficient matrix;
(c) using Cramer’s rule.
See the reverse side for in-class presentation exercises.
The following exercises are for in-class presentation on Wednesday, November 3.
(a) A is called nilpotent if Ak^ = O for some k where O is the n by n zero matrix. Prove that |A| = 0 if A is nilpotent.
(b) A is called lower triangular if the all the entries above the main diagonal are 0. Describe |A| in terms of the entries of A.
(c) C is said to be similar to A if there is an invertible matrix B such that C = B−^1 AB. Prove that |A| = |C| if C is similar to A.
(d) The matrix A is said to be orthogonal if AAt^ = I. Prove that |A| = ±1 if A is an orthogonal matrix.
(e) Suppose A is upper triangular. Prove that A is invertible if and only if all of its diagonal entries are nonzero.