Question

Show that \(\operatorname{det} A B=\operatorname{det} B A\) holds for any two \(n \times n\) matrices \(A\) and \(B\).

Short answer

The property \(\operatorname{det}(AB) = \operatorname{det}(BA)\) holds due to the commutative nature of determinant multiplication.

Step-by-step solution

Recall the property of determinants and matrix products

The determinant of a product of two matrices is equal to the product of their determinants. That is, for any square matrices \(A\) and \(B\), we have \(\operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B)\).

Apply the property to \(AB\) and \(BA\)

Using the property from Step 1, we can write \(\operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B)\) and \(\operatorname{det}(BA) = \operatorname{det}(B) \cdot \operatorname{det}(A)\).

Compare the expressions

Since multiplication of determinants is commutative (i.e., \(\operatorname{det}(A) \cdot \operatorname{det}(B) = \operatorname{det}(B) \cdot \operatorname{det}(A)\)), it follows that \(\operatorname{det}(AB) = \operatorname{det}(BA)\).

Conclude the proof

With the commutative property of scalar multiplication, we conclude that \(\operatorname{det}(AB) = \operatorname{det}(BA)\) holds for any \(n \times n\) matrices \(A\) and \(B\).

Key concepts

Determinants

A determinant is a special number that can be calculated from a square matrix. It provides key information about the properties of the matrix. For an \(n \times n\) matrix, the determinant can tell us whether the matrix has an inverse, whether it scales or rotates space, and what its volume transformation is in higher dimensions.

Calculating a determinant involves summing products, where each product is the multiplication of elements from the matrix. This may sound complex, but for a 2x2 matrix like \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), it's fairly straightforward. You find the determinant using the formula \(ad - bc\).

• If the determinant is zero, the matrix does not have an inverse and is said to be singular.
• A non-zero determinant indicates the matrix is invertible or non-singular.
• Determinants can be used in solving systems of linear equations, finding eigenvalues, and in other operations such as finding areas and volumes.

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix theory, where two matrices are multiplied to produce a third matrix. This operation is not as simple as multiplying numbers; it involves a combination of element-wise multiplications and additions.

To multiply two matrices, multiply the elements of the rows of the first matrix by the elements of the columns of the second matrix and sum the products. This operation requires that the number of columns in the first matrix match the number of rows in the second matrix.

For instance, if you have two matrices, \(A = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}\), the product \(AB\) would be a matrix where each entry is calculated based on the rows from \(A\) and columns from \(B\).

• Matrix multiplication is associative: \((AB)C = A(BC)\).
• Matrix multiplication is distributive: \(A(B + C) = AB + AC\).
• Matrix multiplication is generally not commutative: \(AB eq BA\) in most cases, except in special situations like in our exercise concerning determinants.

Commutative Property

The commutative property is a fundamental principle of mathematics which states that the order of certain operations does not affect the result. For real numbers and simpler operations like addition and multiplication, \(a + b = b + a\) and \(a \cdot b = b \cdot a\) hold true.

However, in matrix theory, multiplication is typically not commutative. This means for most matrices, \(AB eq BA\). There are exceptions though, particularly within the realm of determinants. As seen in the original exercise, while \(AB\) and \(BA\) are normally different, their determinants are the same due to the commutative nature of multiplying scalar values.

In this context, some important points to note are:

• For scalars, the property holds for both addition and multiplication.
• For matrices, generally only addition is commutative.
• The exception of determinant commutativity illustrates the nuanced behaviors within linear algebra.

Understanding these nuances is key in mastering matrix operations and harnessing their power in applied mathematics, data science, and engineering applications.