Question

Problem \(17(\mathrm{b})\) is a special case of the general theorem that the inverse of a product of matrices is the product of the inverses in reverse order. Prove this. Hint: Multiply ABCD times \(\mathrm{D}^{-1} \mathrm{C}^{-1} \mathrm{B}^{-1} \mathrm{A}^{-1}\) to show that you get a unit matrix.

Short answer

The inverse of ABCD is \(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1}\).

Step-by-step solution

Understand the Problem

The aim is to prove that the inverse of a product of matrices is the product of the inverses in reverse order. Specifically, we need to show that the inverse of the product of matrices ABCD is \(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1}\).

Write Down the Given Product and Its Inverse

Start with the product of matrices ABCD. To prove the theorem, multiply ABCD by the claimed inverse matrix \(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1}\).

Perform the Multiplication

Calculate the product: \((ABCD)(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1})\). To prove these are inverses, the result should be the identity matrix. Let's do the multiplication step-by-step:

Use Associative Property of Matrices

Use the associative property of matrix multiplication: \[((ABCD)(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1})) = A (B (C (D \text{D}^{-1}) \text{C}^{-1}) \text{B}^{-1}) \text{A}^{-1}\].

Simplify the Product

Simplify each pair of matrix and its inverse: \((D \text{D}^{-1}) = I, (C \(I\) \text{C}^{-1}) = (C \text{C}^{-1}) = I, (B \(I\) \text{B}^{-1}) = (B \text{B}^{-1}) = I, (A \(I\) \text{A}^{-1}) = (A \text{A}^{-1}) = I\). Finally, \(( I \(I\) \(I\) \(I\)) = I\). So, \((ABCD)(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1}) = I\).

Conclude the Proof

Since \((ABCD)(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1}) = I\), we have shown that the inverse of the product of matrices ABCD is indeed \(\text{D}^{-1} \text{C}^{-1} \text{B}^{-1} \text{A}^{-1}\), which proves the theorem.

Key concepts

matrix multiplication

Matrix multiplication is fundamental in linear algebra and has unique rules. It involves multiplying rows of the first matrix by columns of the second matrix and summing the products.
If matrix A is of size m×n and matrix B is of size n×p, their product AB will be an m×p matrix.

Key points to remember:

• The number of columns in the first matrix must equal the number of rows in the second.
• Matrix multiplication is not commutative; AB ≠ BA in most cases.
• The product AB is obtained by summing the products of corresponding elements from rows of A with columns of B.

Practicing multiplication with smaller matrices can help solidify the concept.

associative property

The associative property is vital when working with matrix multiplication. It states that for any matrices A, B, and C, the equation (AB)C = A(BC) holds true.
This property allows us to group matrices in different ways without changing the product’s result.

In practical terms:

• It simplifies complex multiplications.
• We can re-arrange matrices for easier computation without altering the outcome.
• This is particularly useful in proofs and larger calculations like in the current exercise.

For example, in the proof of the inverse product of matrices, the associative property is used multiple times to re-group matrices and simplify the proof.

identity matrix

The identity matrix, denoted as I, acts like the number 1 in matrix multiplication. Multiplying any matrix by I leaves the original matrix unchanged.
An identity matrix is a square matrix (same number of rows and columns) with 1s on the diagonal and 0s elsewhere.

Key points:

• Multiplying a matrix A by the identity matrix I (on either side) results in A: AI = IA = A.
• The identity property is integral for proving relationships involving matrix inverses.
• For example, ID = D, DI = D for any matrix D of the same size.

Understanding the identity matrix is essential for comprehending more complex matrix operations and proofs.

matrix inverses

A matrix inverse is a matrix that, when multiplied by the original matrix, results in the identity matrix. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to have an inverse.
If A is a matrix, its inverse is denoted as A^-1.

Important aspects:

• A matrix A multiplied by its inverse A^-1 yields the identity matrix: AA^-1 = A^-1A = I.
• Inverses are used to solve linear equations and prove matrix-related theorems.
• For a product of matrices, the inverse of the product is the product of the inverses in reverse order: (AB)^-1 = B^-1A^-1.

Understanding and working with matrix inverses is crucial for solving complex linear algebra problems like the one in the exercise.