MTH 210 Study Guide - Midterm Guide: Row Echelon Form, Elementary Matrix, Laplace Expansion
27 Nov 2017
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2.1: Determinants by Cofactor Expansion
• Define Determinant
• How to find determinant of 2x2 matrix
• Define determinant for an nxn matrix
• Give example of finding determinant by cofactor expansion for a 4x4 matrix
• Define minor of entry aij
• Define cofactor of entry aij
• Explain determinant of a triangular matrix
• Explain why determinant of transpose of A is determinant of A
• T/F
o The number obtained by a cofactor expansion of matrix A is independent of the
row or column chosen for the expansion-T
o If A is a square matrix whose minors are all 0, then det(A)=0-T
o For every square matrix A & every scalar c, is it true that det(cA)=cdet(A)-F
o For all square matrices A &B is it true that det(A+B)=det(A)+det(B)-F
2.2: Evaluating Determinants by Row Reduction
• Theorem on switching two rows
• Theorem on multiplying by nonzero constant
• Theorem on adding a multiple of one row to another
• Theorem on determinants of elementary matrices
• Give example of using row reduction to obtain determinant
• Give theorem on two identical rows or columns
• Give theorem on row that is multiple of another row
• Give theorem on row or column of zeroes
• How to do row reduction if columns are nicer
• Theorem on if matrix B is obtained from A by an elementary row operation
• Theorem on if A is row equivalent to B
• Theorem on reduced row echelon form of A
• Give EST
• T/F
o If A is a 4x4 matrix & B is obtained from A by interchanging the first two rows
and then interchanging the last two rows, det(B)=det(A)-T
o If A is a 3x3 matrix & B is obtained from A by multiplying the first column by 4 &
multiplying the 3rd column by ¾, than det(B)=3det(A)-T
o If A is a square matrix with two identical columns, then det(A)=0-T
o If the sum of the second & fourth row vectors of a 6x6 matrix A is equal to the
last row vector, then det(A)=0-T
2.3: Properties of Determinants
• Give theorem on det(kA) when k is a scalar
• Give theorem on det(AB)
• Prove: If E is an elementary nxn matrix & B is any nxn matrix, then det(EB)=det(E)det(B)
• Find det− from det(A)
• Define the matrix of cofactors
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
If a is a square matrix whose minors are all 0, then det(a)=0-t: for every square matrix a & every scalar c, is it true that det(ca)=cdet(a)-f, for all square matrices a &b is it true that det(a+b)=det(a)+det(b)-f. If a is a 4x4 matrix & b is obtained from a by interchanging the first two rows and then interchanging the last two rows, det(b)=det(a)-t. If a is a 3x3 matrix & b is obtained from a by multiplying the first column by 4 & multiplying the 3rd column by , than det(b)=3det(a)-t. If a is a square matrix with two identical columns, then det(a)=0-t. If the sum of the second & fourth row vectors of a 6x6 matrix a is equal to the last row vector, then det(a)=0-t. If a is a square matrix & the linear system ax=0 has multiple solutions for x, then det(a)=0-t. If a is a 3x3 matrix, then det(2a)=2det(a)-f.
