031 Review Part 2, Problem 5

Let  $A$  and  $B$  be  $6\times 6$  matrices with  ${\text{det }}A=-10$  and  ${\text{det }}B=5.$  Use properties of determinants to compute:

(a)  ${\text{det }}3A$

(b)  ${\text{det }}(A^{T}B^{-1})$

Foundations:
Recall:
1. If the matrix  $B$  is identical to the matrix  $A$  except the entries in one of the rows of  $B$
are each equal to the corresponding entries of  $A$  multiplied by the same scalar  $c,$  then
${\text{det }}B=c({\text{det }}A).$
2.  ${\text{det }}(AB)=({\text{det }}A)({\text{det }}B)$
3. For an invertible matrix  $A,$  since  $AA^{-1}=I$  and  ${\text{det }}I=1,$  we have
${\text{det }}A^{-1}={\frac {1}{{\text{det }}A}}.$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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