# Easy matrix/determinants question

• astonmartin
In summary, using the given information, we can find that det C = y/(8x), which is not one of the solutions given.
astonmartin

## Homework Statement

Suppose A and B are 3 x 3 matrices and det A = x ≠ 0 while det B = y. Let
C be the matrix ((2A)^-1 )B <-- (2A) inverse x B

then det C is:

## The Attempt at a Solution

det(2A) = 2x, so det 2A inverse = 1/(2x)
det C = y/(2x)...which is not one of the solutions

a) y/8x b) 2xy c) -2y/x d) 2y/x e) 8y/x

what am I missing here?

You are very close; however,

$$\det(\alpha A)=\alpha^{n} \det(A)$$

where n is the order of the matrix A, in this case 3. To understand why this happens, think of the determinant of the identity and multiply it by a scalar.

## What is a matrix and how is it used in mathematics?

A matrix is a rectangular array of numbers or symbols that are arranged in rows and columns. It is used in mathematics to represent and solve systems of equations, transformations, and other mathematical operations.

## What is a determinant and what is its significance in matrix operations?

A determinant is a numerical value that is calculated from the elements of a square matrix. It is significant in matrix operations because it can determine whether a matrix has a unique solution, and it is also used to find the inverse of a matrix.

## How do you calculate the determinant of a 2x2 matrix?

The determinant of a 2x2 matrix is calculated by multiplying the elements in the main diagonal and subtracting the product of the elements in the other diagonal. For example, the determinant of the matrix [a b; c d] is ad - bc.

## What is the relationship between the determinant and the area/volume of a shape?

The absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the column vectors of the matrix. Similarly, the absolute value of the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by the column vectors of the matrix.

## Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the arrangement of the elements in the matrix and does not affect its numerical value.

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