What is the Rule for Expanding Determinants?

[~Sam~]

Homework Statement

Let A, B and C be 3 x 3 invertible matrices where det A = -2 ,det B = -2 and det C is some non-zero scalar.

Then det (CTA−1B2C−1) = ?

and det [ −2(A2)TC2B−1(C−1)2] = ?

the T represents transpose and the -1 represents inverse.

Homework Equations

What does the non-zero scalar mean?

The Attempt at a Solution

I used the rule to expand the products of the determinants, but I'm not sure what to do next and what it means by non-zero scalar.

 

[calimechengr]

When you find the determinant of a matrix, your result is a scalar, or a single number. Non zero just means that det(c) is a scalar other than zero.

 

[~Sam~]

When you find the determinant of a matrix, your result is a scalar, or a single number. Non zero just means that det(c) is a scalar other than zero.

So how would it affect my answer since I really don't know the value of det(c).

 

[Mark44]

Do you know any theorems about determinants? One that I remember is that det(A^T) = det(A). You'll need some of these theorems in these problems, particularly one for det(A^-1) as it relates to det(A), and one for det(Aⁿ) as it relates to det(A).

Tip: To make exponents (for transposes and matrix inverses), click the Go Advanced button below the text entry field. This opens a menu of buttons you can use to format what you write. The X² button can be used for exponents and the X₂ button can be used for subscripts.

Here are your problems, formatted for easier reading:
a) det(C^TA⁻¹B²C⁻¹)
b) det( −2(A²)^TC²B⁻¹(C⁻¹)²)

 

[HallsofIvy]

For these problems the most crucial thing you need to know is that det(AB)= det(A)det(B). If you know that, these problems are easy. If you don't, ---.