Problem 9 - College Algebra with Modeling and Visualization

Question

Let \(a_{i j}\) and \(b_{i j}\) be general elements for the given matrices \(A\) and \(B\). (a) Identify \(a_{12}, b_{32},\) and \(b_{22}\) (b) Compute \(a_{11} b_{11}+a_{12} b_{21}+a_{13} b_{31}\) (c) If possible, find a value for \(x\) that makes \(A=B\). $$A=\left[\begin{array}{rrr}1 & 3 & -4 \\\3 & 0 & 7 \\\x & 1 & -1\end{array}\right]$$ $$B=\left[\begin{array}{rrr}1 & x & -4 \\\3 & 0 & 7 \\\3 & 1 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

(a) 3, 1, 0; (b) -2; (c) x = 3.

Step 1: Identify Matrix Elements

First, let's identify the required elements from matrices \(A\) and \(B\). The element \(a_{12}\) refers to the element in the first row and second column of matrix \(A\). From matrix \(A\), \(a_{12} = 3\).Next, \(b_{32}\) refers to the element in the third row and second column of matrix \(B\). From matrix \(B\), \(b_{32} = 1\).Finally, \(b_{22}\) refers to the element in the second row and second column of matrix \(B\). From matrix \(B\), \(b_{22} = 0\).

Step 2: Compute the Expression

We need to compute the expression \(a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31}\). First identify the necessary elements:- \(a_{11} = 1\), \(b_{11} = 1\)- \(a_{12} = 3\), \(b_{21} = 3\)- \(a_{13} = -4\), \(b_{31} = 3\)Now substitute these values into the expression:\[a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31} = (1)(1) + (3)(3) + (-4)(3)= 1 + 9 - 12= -2\]

Step 3: Determine if A Equals B

To check if \(A = B\), each corresponding element in \(A\) and \(B\) must be equal. Comparing each element of matrices:- First row: (1, 3, -4) in \(A\), (1, \(x\), -4) in \(B\)- Second row: (3, 0, 7) in both\(A\) and \(B\)- Third row: (\(x\), 1, -1) in \(A\), (3, 1, -1) in \(B\)From the first row and third row, we get the equations:\(3 = x\) and \(x = 3\).Since \(x\) must satisfy both, \(x = 3\).

Key Concepts

Matrix Element Identification

When working with matrices, identifying matrix elements is one of the first steps we must understand.
Each element in a matrix is located by its row and column position.
For example, an element denoted as \(a_{ij}\) refers to the element at the intersection of the \(i\)-th row and \(j\)-th column of a matrix \(A\). Understanding this is crucial for extracting or working with specific data contained in matrices.

In the given exercise, we are asked to identify specific elements. Let's break it down:

• \(a_{12}\) is the element from the first row and the second column of matrix \(A\), noted as 3 in our example.
• \(b_{32}\) is the element from the third row and second column of matrix \(B\), which is 1 according to the given matrix.
• \(b_{22}\) is the element from the second row and second column of matrix \(B\), found to be 0 in this case.

Understanding these locations allows us to perform further calculations with accuracy, which is fundamental in matrix operations.

Matrix Equality

Matrix equality is a simple yet powerful concept.
Two matrices are equal if and only if they have the same dimensions and each corresponding element is identical.
This means that every element from the first row to the last must match exactly with its counterpart in another matrix.

In our exercise, to establish *Matrix Equality* for matrices \(A\) and \(B\), we checked each element in both matrices:

• First row - \(A\) contains elements \( (1, 3, -4) \), while \(B\) has \( (1, x, -4) \). This gives us an equation where \(x\) must equal 3 for the equality to hold.
• Second row - Both matrices have identical elements \((3, 0, 7)\).
• Third row - Comparing \((x, 1, -1)\) in \(A\) and \((3, 1, -1)\) in \(B\), reconfirms \(x\) must be 3.

Thus, solving for the value of \(x\) ensures that all elements match and the matrices are indeed equal.
Matrix equality ensures coherence and is used widely to verify solutions or solve for unknown elements within matrices.

Matrix Multiplication

Matrix multiplication allows us to derive new information from existing matrices.
It's not as straightforward as regular arithmetic multiplication since the process involves a sum of products.
In this process, the number of columns in the first matrix must match the number of rows in the second matrix.

• Consider an expression where we need to multiply two matrices to compute specific elements as in our exercise: \(a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31}\).
• Identify the required elements: \(a_{11} = 1\), \(b_{11} = 1\), \(a_{12} = 3\), \(b_{21} = 3\), \(a_{13} = -4\), and \(b_{31} = 3\).
• Substitute these into the equation and simplify: \[1 \times 1 + 3 \times 3 + (-4) \times 3 = 1 + 9 - 12 = -2\]

Matrix multiplication's primary usage is combining and transforming matrices to acquire new insights or solve complex problems efficiently.
Understanding this process is a key aspect of linear algebra.