Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -2 \end{array}\right], \quad B=\left[\begin{array}{rrr} \frac{1}{3} & 0 & 0 \\ 0 & -\frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{6} \end{array}\right]$$

Step-by-Step Solution

Answer

The product of matrices \(A\) and \(B\), denoted as \(AB\), is an Identity Matrix, i.e., \( \left[\begin{array}{rrr} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right]\)

Step 1: Multiply Corresponding Elements

Begin by multiplying corresponding elements in the two matrices. For a diagonal matrix, we only need to consider the diagonal elements since the other elements are all zero. That is, multiply first element of matrix \(A\) (\(6\)) with the first element of matrix \(B\) (\(\frac{1}{3}\)), second element of \(A\) (\(4\)) with the second element of \(B\) ( -\(\frac{1}{4}\)), and third element of \(A\) (-\(2\)) with the third element of \(B\) (\(\frac{1}{6}\)).

Step 2: Write Down the Results

Once the multiplication is done, write down the results to the corresponding positions in the new matrix, \(AB\). All the remaining positions in the new matrix that do not have corresponding elements in \(A\) and \(B\) (non-diagonal elements) should be zero, since they are obtained by multiplying non-diagonal elements of \(A\) and \(B\) which are all zero.

Step 3: Verify the Identity Matrix

The result from the multiplication should be an identity matrix. An identity matrix is a square matrix in which all the main diagonal elements are \(1\) and all other elements are \(0\). Thus matrix \(AB\) should be an identity matrix.

Key Concepts

Diagonal Matrix

A diagonal matrix is an essential concept in linear algebra, characterized by all its non-diagonal elements being zero. This means that only the numbers located on the diagonal line that stretches from the top-left corner to the bottom-right corner of the matrix are non-zero, typically represented as

• A matrix where the diagonal might have any number, while all other spots in the matrix are zero.
• The general form can be written as: \[\begin{bmatrix} a_{11} & 0 & 0 \ 0 & a_{22} & 0 \ 0 & 0 & a_{33} \end{bmatrix}\]
• In diagonal matrices, those diagonal entries (e.g., \(a_{11}, a_{22}, a_{33}\)) are the only relevant numbers when performing arithmetic operations such as addition or multiplication with other diagonal matrices.

In the exercise provided, matrix \(A\) is a diagonal matrix\! This greatly simplifies multiplication with another diagonal matrix like \(B\), since only the diagonal elements are multiplied together.

Identity Matrix

An identity matrix is a special type of diagonal matrix where the main diagonal elements are all precisely \(1\), and every other element is \(0\). This is crucial in linear algebra because it acts like the number \(1\) in matrix multiplication; multiplying any matrix by an identity matrix of appropriate size leaves the original matrix unchanged.

• Consider an identity matrix as: \[\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
• In matrix multiplication, the identity matrix functions as a multiplicative identity, meaning for any square matrix \(C\), if \(I\) is the identity matrix, then \(IC = CI = C\).
• The resulting matrix \(AB\) in the exercise is intended to be an identity matrix, highlighting how perfectly aligned the diagonal elements from \(A\) and \(B\) should be to achieve this result.

Understanding the identity matrix helps appreciate why certain matrices remain unchanged after multiplication, which enhances problem-solving strategies in matrix operations.

Matrix Element Wise Multiplication

Element-wise multiplication, also sometimes called the Hadamard product, is a straightforward approach where corresponding elements of two matrices are multiplied to create a new matrix. This is different from traditional matrix multiplication where rows and columns are computed together.

• In essence, this involves multiplying each position (i.e., each element) of one matrix by the corresponding position of another matrix.
• For example, if you have two matrices: \(X =\begin{bmatrix} x_{11} & x_{12} \ x_{21} & x_{22} \end{bmatrix}\) and \(Y =\begin{bmatrix} y_{11} & y_{12} \ y_{21} & y_{22} \end{bmatrix}\), their element-wise multiplication results in: \[ \begin{bmatrix} x_{11}y_{11} & x_{12}y_{12} \ x_{21}y_{21} & x_{22}y_{22} \end{bmatrix} \]
• Unlike matrix multiplication, element-wise multiplication requires that the matrices are of the same size.

In the context of our exercise, even though the multiplication of matrices \(A\) and \(B\) wasn't element-wise, understanding this concept clarifies how entries in specific positions contribute directly to the final matrix product when focusing purely on corresponding diagonal elements.