Problem 93 - Precalculus

Question

Determine whether the statements are true or false. $$\begin{array}{l}\text { If } A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22} \end{array}\right] \text { and } B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right], \text { then } \\\A B=\left[\begin{array}{ll}a_{11} b_{11} & a_{12} b_{12} \\\a_{21} b_{21} & a_{22} b_{22}\end{array}\right]\end{array}$$

Step-by-Step Solution

Answer

The statement is false.

Step 1: Understand Matrix Multiplication

Matrix multiplication is not conducted by multiplying corresponding elements. For two matrices, \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \), their product \( AB \) is calculated as follows: each element of the resulting matrix is a dot product of rows of the first matrix and columns of the second matrix.

Step 2: Apply the Correct Formula

The elements of the product matrix \( AB \) are calculated as: \( AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} \).

Step 3: Compare with Statement

Compare the matrix formed by applying the correct formula \( AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} \) with the matrix given in the statement \( \begin{bmatrix} a_{11} b_{11} & a_{12} b_{12} \ a_{21} b_{21} & a_{22} b_{22} \end{bmatrix} \).

Step 4: Determine True or False

The given matrix in the statement is not equal to the matrix formed by the formula for matrix multiplication. So, the statement is false.

Key Concepts

Matrix Product

Matrix multiplication, often referred to as the matrix product, is a core concept in linear algebra. It involves two matrices and produces another matrix by performing specific operations. Contrary to simple element-wise multiplication, the matrix product requires a specific method to compute the resultant matrix:

• Consider two matrices, \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \).
• The process involves taking the dot product of rows from the first matrix with the columns of the second matrix.
• Resultant elements in the new matrix, say matrix \( AB \), are derived from these dot products.

Thus, each element of the product matrix \( AB \) is calculated using all elements in a row multiplied by corresponding elements in a column then summed together, not merely multiplying corresponding terms.

Dot Product

The central operation within the matrix product calculation is the dot product. This is a specific way of combining two sets of numbers, most commonly vectors, to produce a single number:

• The dot product between two vectors involves multiplying corresponding components and then summing those products.
• For finding a matrix product, each element of the resulting matrix is a dot product of a row from the first matrix and a column from the second matrix.
• For example, to compute the element at the first row and first column of the matrix \( AB \), compute the dot product of the first row of \( A \) with the first column of \( B \): \( a_{11}b_{11} + a_{12}b_{21} \).

Remember, the dot product outputs a scalar, which then fills the appropriate position in the resultant matrix.

Matrix Equality

Matrix equality is a concept that explains when two matrices are considered equal. Here’s how you determine matrix equality:

• Two matrices are equal if they have the same dimensions.
• More importantly, all corresponding elements in both matrices must be identical.
• In the context of matrix multiplication results, after computing the product matrix, compare it element by element with another matrix to check for equality.

In the given exercise, matrix equality is crucial in determining whether a hypothesized matrix product statement is correct. If the product matrix differs in any entry from another matrix purported to be equal, the statement is false.