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Reason - Linear Algebra - Quiz, Exercises of Linear Algebra

This is the Quiz of Linear Algebra which includes Satisfies the Conditions, Information, Explicit Conditions, Guarantee, Conditions, Basis, Linear Combination, Basis Vectors, Column Vector, Bottom Entry etc. Key important points are: Reason, Determinant, Matrix, Swapping Rows, Derivative, Integral, Elementary Row Operations, Main Diagonal Entries, Multiplied, Triangular Matrix

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

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Math 205 A B Quiz 06 gold page 1 11/02/2012 Name
1. Find the determinant of the following matrix by hand, showing all your steps (intermediate results) along the way.
Make good use of 0’s. Circle your final answer.
2 4 0 5
5 0 0 11
12 8 3 7
9 0 0 6
2. Let B=
1 2 4
a b c
r s t
; suppose det(B)=5
Find the determinant of each of the following matrices, and under each matrix write the reason/rule/fact about determinants
of matrices you used to find the det. (eg, “swapping rows changes the sign of the det” or “the determinant of the derivative
of a matrix is the matrix of its integral” (this second fact is nonsense)
2a) M=
1a r
2b s
4c t
2b) N=
1 2 4
a b c
r+ 2 s+ 4 t+ 8
2c) Q=
10 20 40
a b c
r s t
2d) R=
10 20 40
a b c
124
2e) S= 4B
3) Suppose the following elementary row operations turn the matrix Ainto U:
First, rows r1and r2are swapped. Second, r3r3+ 10r2, Third, row 1 is multiplied by 4. The matrix Uis a 3 ×3 upper
triangular matrix, with main diagonal entries 4, 3, and 2.
3A) What is det(A)? Explain.
3B) Is Ainvertible? Explain how you know.

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Math 205 A B Quiz 06 gold page 1 11/02/2012 Name

  1. Find the determinant of the following matrix by hand, showing all your steps (intermediate results) along the way. Make good use of 0’s. Circle your final answer. 

 

  1. Let B =

a b c r s t

; suppose det(B)=

Find the determinant of each of the following matrices, and under each matrix write the reason/rule/fact about determinants of matrices you used to find the det. (eg, “swapping rows changes the sign of the det” or “the determinant of the derivative of a matrix is the matrix of its integral” (this second fact is nonsense)

2a) M =

1 a r 2 b s 4 c t

 (^) 2b) N =

a b c r + 2 s + 4 t + 8

2c) Q =

a b c r s t

 (^) 2d) R =

a b c 1 2 4

2e) S = 4B

  1. Suppose the following elementary row operations turn the matrix A into U : First, rows r 1 and r 2 are swapped. Second, r 3 ← r 3 + 10r 2 , Third, row 1 is multiplied by 4. The matrix U is a 3 × 3 upper triangular matrix, with main diagonal entries 4, 3, and 2. 3A) What is det(A)? Explain.

3B) Is A invertible? Explain how you know.