Question

If \(A B=B A\), then prove that \(A B A B=A^{2} B^{2} .\) The following are the steps involved in proving the above result. Arrange them in the sequential order. (a) \(A B A B=A(B A) B\) (b) \((\mathrm{AA})(\mathrm{BB})\) (c) \(\mathrm{A}(\mathrm{AB}) \mathrm{B}\) (d) \(\mathrm{A}^{2} \mathrm{~B}^{2}\) (1) abcd (2) acbd (3) bcad (4) adbc

Short answer

Short Answer: If given that AB = BA, to prove that ABAB = A^2B^2, follow these steps: 1. Apply the given condition: ABAB = A(BA)B (using AB = BA) 2. Use associativity property of matrix multiplication: ABAB = A(AB)B 3. Apply the given condition again: ABAB = (AA)(BB) (using AB = BA) 4. Simplify: ABAB = A^2B^2 The correct sequence is acbd.

Step-by-step solution

(1) Apply the given condition \(AB = BA\)

Starting with the given condition, we are going to find the expression for \(ABAB\). Observe the following: \(ABAB = A(BA)B\) (using \(AB = BA\))

(2) Use associativity property of matrix multiplication

Since matrix multiplication is associative, we can write the above expression as: \(ABAB = A(AB)B\)

(3) Apply the given condition \(AB = BA\) again

Now, using the given condition \(AB = BA\) again, we can write the above expression as: \(ABAB = (AA)(BB)\)

(4) Simplify using powers

Finally, we can represent the above expression using powers of A and B matrices as: \(ABAB = A^2B^2\) Hence, if \(AB = BA\), then \(ABAB = A^2B^2\). The correct sequence of steps is represented by the option (2) acbd.

Key concepts

Associative Property

In matrix multiplication, the associative property is a fundamental rule that simplifies complex matrix expressions. It states that when multiplying three or more matrices together, the grouping of matrices does not affect the product. This means that

• If you have matrices A, B, and C, then
• \((AB)C = A(BC)\).

This property was crucial in our exercise for rearranging matrix terms without changing the outcome. It helped us simplify the expression \(ABAB\) to \(A(AB)B\) so that we could further break it down using the given condition \(AB=BA\).
This flexibility of grouping allows us to focus more on rearranging and aligning matrices according to our needs without worrying about how they're grouped, which can drastically reduce computational complexity and error probability while solving.

Matrix Powers

Matrix powers involve multiplying a matrix by itself multiple times. If a matrix A is multiplied by itself n times, it is denoted as \(A^n\). For the given proof, matrix powers help in condensing repeated multiplication into a more elegant and manageable expression.
By understanding matrix powers:

• We simplified \((AA)(BB)\) into \(A^2B^2\) in the solution.
• It enabled us to clearly see the transformation of the matrix expressions when the matrices commute.

This simplification not only aids in mathematical elegance but also eases computation, as calculating powers of matrices can be more straightforward than performing each multiplication separately.

Matrix Equality

Matrix equality is a crucial concept that dictates when two matrices are considered the same. Two matrices, say A and B, are equal if each corresponding element in the matrices is equal.
In our exercise, this principle sums up in the condition \(AB=BA\). This implies that matrices A and B commute and that any rearrangement still results in matrix equality. Using this property:

• We confirmed that \(ABAB\) was indeed equal to \(A^2B^2\).
• It reassured us that rearrangements using the associative property were valid steps.

Understanding this equality is foundational in proving more complex matrix relationships and confirms that operations done on matrices maintain their equivalences throughout.

Commutative Property

Unlike numbers, matrices usually do not commute. Generally, for two matrices A and B, \(AB eq BA\). The commutative property is a property that, if and when it holds, allows the switching of matrices in multiplication.In the above exercise, it is given that \(AB = BA\), which is a rare condition. This property:

• Allowed us to substitute \(AB\) with \(BA\) interchangeably without affecting the expression.
• Helped turn \(ABAB\) into \((AA)(BB)\), setting up the use of powers, \(A^2B^2\).

The commutative property is generally not valid for matrices, so when a problem states matrices commute, it opens doors to simplify and solve in ways not usually possible.