Linear Algebra Review 1
Solutions
Chapter 3 Systems of Equations
3.0 Linear Algebra Review
In Exercises 1 – 9, compute the indicated matrices given
A=1−1 3
205, B =
2 1 0
−3−1 5
1 3 4
, C =
4 2
3−1
2−4
and D=
1−1 4
0 2 −2
0 0 3
.
If an operation cannot be performed, indicate why not.
1. (a) 2A+CT
(b) C−3B
(a)
2. (a) AB
(b) AD
(a)
3. (a) CA
(b) AC
(a)
4. (a) BD
(b) DB
(a)
5. (a) BC
(b) CB
(a)
6. (a) 3B−2D
Linear Algebra Review 3
(b) 2DT+B
(a)
7. (a) det(D)
(b) det(A)
(a) Expanding along the first column, we find
8. (a) CTD
(b) BAT
(a)
9. (a) −2AT+ 5C
(b) BT+D
(a)
4Section 3.0
(b)
10. Let Abe a nonsingular matrix.
(a) Show that A−1is unique.
(b) Show that A−1is nonsingular and (A−1)−1=A.
(c) Show that ATis nonsingular and (AT)−1= (A−1)T.
(d) If Bis nonsingular, show that AB is nonsingular and (AB)−1=B−1A−1.
(a) Let Abe a nonsingular matrix, and suppose, for sake of contradiction, that
(b) Let Abe a nonsingular matrix and consider the matrix B=A−1. Because
(c) Let Abe a nonsingular matrix. Because
(d) Let Aand Bbe nonsingular matrices. Because
11. Can an n×mmatrix with n6=mbe symmetric? Explain.
12. Recalculate the determinant of the matrix
A=
1 0 4 1
−2 1 −3 2
0 0 0 2
321−1
by first expanding along the second column.
13. Show that
det a11 a12
a21 a22 =a11a22 −a12a21.
14. Let
A=a11 a12
a21 a22 .
(a) Show that Ais nonsingular provided a11a22 −a12a21 6= 0.
(b) If a11a22 −a12 a21 6= 0, show that
A−1=1
a11a22 −a12 a21 a22 −a12
−a21 a11 .
6Section 3.0
15. Let Dbe an n×ndiagonal matrix. Show that det(D) = d11d22d33 ···dnn.
Let
d11
Repeatedly expanding along the first row leads to
Linear Algebra Review 7
d44
16. Let αbe a real number and let
A=α4
1αand B=
2α0
−3−1 5
1 3 α
.
(a) For what value(s) of αis Asingular?
(b) For what value(s) of αis Bsingular?