Guided Proof Prove that if A and B are diagonal matrices (of...

Guided Proof Prove that if $A$ and $B$ are diagonal matrices (of the same size), then $A B=B A$ Getting Started: To prove that the matrices $A B$ and $B A$ are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be two diagonal $n \times n$ matrices. (ii) The ijth entry of the product $A B$ is $c_{i j}=\sum_{k=1}^{n} a_{i k} b_{k j}$ (iii) Evaluate the entries $c_{i j}$ for the two cases $i \neq j$ and $i=j$ (iv) Repeat this analysis for the product $B A$

Guided Proof Prove that if $A$ and $B$ are diagonal matrices (of the same size), then $A B=B A$

Getting Started: To prove that the matrices $A B$ and $B A$ are equal, you need to show that their corresponding entries are equal.
(i) Begin your proof by letting $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be two diagonal $n \times n$ matrices.
(ii) The ijth entry of the product $A B$ is $c_{i j}=\sum_{k=1}^{n} a_{i k} b_{k j}$
(iii) Evaluate the entries $c_{i j}$ for the two cases $i \neq j$ and $i=j$
(iv) Repeat this analysis for the product $B A$

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Key Concepts

Diagonal Matrices

A diagonal matrix is a square matrix in which all entries outside the main diagonal are zero. This property simplifies many operations since only the diagonal elements contribute to the matrix's behavior, especially during multiplication.

Matrix Multiplication

Matrix multiplication is defined by computing each entry of the product as the sum of the products of corresponding entries from the row of the first matrix and the column of the second. This process relies on summing over indices and is greatly simplified when one or both matrices have many zero entries.

Commutativity in Special Cases

Although matrix multiplication is generally noncommutative, certain matrices, like diagonal matrices, commute because their structure forces most terms in the product to vanish, leaving only products of the diagonal elements. This property is a special case that emphasizes how structure can lead to commutativity.

Entry-wise Analysis

Analyzing a matrix product entry by entry involves isolating the sum that defines each element of the resulting matrix. For diagonal matrices, this analysis shows that non-diagonal entries are zero and diagonal entries are the product of corresponding elements, demonstrating that the order of multiplication does not affect the outcome.

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