Thread: Need help with: Making a Matrix Singular and Finding Determinates

1. Need help with: Making a Matrix Singular and Finding Determinates

Ok, i have been working on these problems for a bout an hour now and cant seem to find an answer.

1.)

Find all possible values of c that will make the matrix singluar

(c-1) -2 -3
(c-1) (c+1) -3
0 0 1

i used the formula (a11)(a22)(a33)-(a11)(a32)(a23)-(a12)(a21)(a33)+(a12)(a31)(a23)+(a13)(a21)(a32)-(a13)(a31)(a22)=0

solved it out to get c^2+2c-3=0
thus my roots are 3 and -1(but thats not the correct answer....)

2.)

Let A and B be 3X3 Matrices with det(A)=2 and det(B)=7

i know that for det(AB) you multiply the 2 but i need to find

det(2A)=?
det(3AB)=?
det(AB^-1)=?

3.)

If the determinate of a 4X4 matrix A is det(A)=9, and the matrix D is obtained from A by adding 3 times the third row to the second, determine det(D).

Any help is greatly appreciated
Thank you,
Drew

2. Hmm. For 1.), I get the determinant to be

$1((c+1)(c-1)-(-2)(c-1))=c^{2}-1+2c-2=c^{2}+2c-3=0,$ the same as you. However, the solutions are $c=1,-3,$ the opposite signs you got.

For 2.), note that for an $n\times n$ matrices $A,B,$ and scalar $a,$ you have

$\det(aA)=a^{n}\det(A),$

$\det(AB)=\det(A)\,\det(B),$

and, if $A$ is invertible, then

$\det(A^{-1})=\dfrac{1}{\det(A)}.$

That should enable you to compute all the required quantities.

3. Thank you, Can anyone explain #3 for me please?

4. Originally Posted by Diggidy
[snip]
3.)

If the determinate of a 4X4 matrix A is det(A)=9, and the matrix D is obtained from A by adding 3 times the third row to the second, determine det(D).

Any help is greatly appreciated
Thank you,
Drew
There is a theorem you need to review regarding what happents to a determinant when you do this sort of elementary row operation. (Your class notes or textbook should list the various theorems for what happens to a determinant for diferent sorts of elementary row operations ....)