# Solving Determinant Question for 3A2B-1

• Dell
In summary: You have summarized the conversation perfectly. In summary, the conversation was about finding the determinant of a given matrix expression and using various operations to simplify it. The final answer was found to be -27/15, with the key steps being multiplying certain rows or columns by -1 to manipulate the matrix and using the given information to solve for the determinant of one of the matrices.
Dell
i am given 2 matrices

A=
a1 b1 c1
a2 b2 c1
a3 b3 c3

B=
-c2 3c1 -c3
b2 -3b1 b3
-5a2 15a1 -5a3

and also given is: 5detA+detB=10

what i need to fin is det(3A2B-1)

what i did to help me was

det(3)=27
det(A2)=detA*detA
det(B-1)=1/detB

i see that if i perform 2 "swaps" on B, once between R1 and R3 , and then between C1 and C2, these actions won't change detB.
now i can divide my new R1 by 5, and my new c1 by 3 and transpose B to get a new B* , detB=(5*3)detB*

now my new matrix B* is ALMOST identical to A, except for the (-) signs before some of its numbers, which are drivin me mad,
how can i get rid of them, if i can get rid of them i can find detA.

any ideas? have all my steps beeen legal??

a1 -a2 -a3
-b1 b2 b3
c1 -c2 -c3

Find "patterns" in the negative signs so you can multiply certain rows or columns by -1 to get rid of them.

thanks, got it.. so...

B*=
a1 -b1 c1
-a2 b2 -c2
-a3 b3 -c3

so i'll multiply det(B*)*(-1)*(-1)*(-1)=detA
so detB* =-detA

detB=(5*3)detB*
detB=-15detA

5detA+detB=10
5detA-15detA=10
detA=-1

now to solve the question
det(3A^2B^-1)=27*(-1)*(-1)*(1/-15)
=-27/15

does this all look right??

Dell said:
thanks, got it.. so...

B*=
a1 -b1 c1
-a2 b2 -c2
-a3 b3 -c3

so i'll multiply det(B*)*(-1)*(-1)*(-1)=detA
so detB* =-detA

detB=(5*3)detB*
detB=-15detA5detA+detB=10
5detA-15detA=10
detA=-1

now to solve the question
det(3A^2B^-1)=27*(-1)*(-1)*(1/-15)
=-27/15

does this all look right??
Right up until to the very end! Since det(A)=-1 and det(B) = -15det(A), we have det(B) = +15.

Apart from that - well done, Dell!

## 1. What is a determinant?

A determinant is a mathematical value that can be calculated for a square matrix. It represents the scaling factor of the transformation described by the matrix.

## 2. How do you solve a determinant question for 3A2B-1?

To solve a determinant question for 3A2B-1, you will need to first expand the expression using the rules of matrix multiplication. Then, you can use properties of determinants, such as row operations and factorization, to simplify the expression and find the final value.

## 3. What are the properties of determinants?

Some properties of determinants include:

• The determinant of a matrix multiplied by a scalar is equal to the scalar multiplied by the determinant of the matrix.
• The determinant of the identity matrix is 1.
• The determinant of a matrix with two identical rows or columns is 0.
• The determinant of a matrix with a row or column of all 0's is 0.

## 4. What are the applications of determinants?

Determinants have many applications in mathematics, physics, and engineering. They are commonly used in solving systems of equations, calculating areas and volumes, and finding inverse matrices. They also have applications in quantum mechanics, computer graphics, and statistics.

## 5. Can determinants be calculated for non-square matrices?

No, determinants can only be calculated for square matrices. A matrix must have the same number of rows and columns in order to have a determinant value.

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