Problem 7 - Precalculus: Mathematics for Calculus

Question

Find the values of \(a\) and \(b\) that make the matrices \(A\) and \(B\) equal. $$A=\left[\begin{array}{rr} 3 & 4 \\ -1 & a \end{array}\right] \quad B=\left[\begin{array}{rr} b & 4 \\ -1 & -5 \end{array}\right]$$

Step-by-Step Solution

Answer

Values are \( a = -5 \) and \( b = 3 \).

Step 1: Understand Equality of Matrices

For two matrices to be equal, all corresponding elements must be the same. Therefore, if matrices \( A \) and \( B \) are equal, \( A_{11} = B_{11} \), \( A_{12} = B_{12} \), \( A_{21} = B_{21} \), and \( A_{22} = B_{22} \).

Step 2: Set Up Equations

From the matrices given, we compare corresponding elements: - \( 3 = b \) from \( A_{11} = B_{11} \). - \( 4 = 4 \) checks out automatically.- \( -1 = -1 \) also checks out.- \( a = -5 \) from \( A_{22} = B_{22} \).

Step 3: Solve Equations

Solve each equation:1. From \( 3 = b \), we get \( b = 3 \).2. From \( a = -5 \), we simply confirm \( a = -5 \).

Step 4: Verify Solution

Check if all corresponding elements now match with these values:- For the first element: \( 3 = b \), thus \( b = 3 \).- For the last element: \( a = -5 \), agrees.All conditions are satisfied, confirming the solution.

Key Concepts

Matrices

A matrix is a rectangular arrangement of numbers, symbols, or expressions, organized in rows and columns. Imagine a spreadsheet with rows and columns filled with numbers — that's what a matrix looks like.
Matrices are widely used in various fields, such as physics, computer graphics, and economics, due to their ability to simplify complex problems. To define a matrix, you state the number of rows and columns it has, usually noted as "m x n" ("m" rows and "n" columns).

• The numbers inside a matrix are called "elements" or "entries."
• Each element of a matrix can be addressed with its position, such as the first row and first column being denoted as the element \( A_{11} \).

Understanding matrices is essential for solving many mathematical problems, like those involving systems of equations. They serve as the foundation upon which more advanced concepts are built.

Element Comparison

To determine whether two matrices are equal, we must compare their corresponding elements. In mathematical terms, two matrices \(A\) and \(B\) are considered equal, denoted as \(A = B\), if and only if every pair of corresponding elements in the matrices are equal. Here's how you can do that:

• Check each element at the same position in both matrices.
• For instance, the element in the first row and first column of matrix \(A\), noted as \( A_{11} \), must be exactly the same as the element in matrix \(B\) at the same position, or \( B_{11} \).

For the exercise example, a mismatch between any pair of these corresponding elements means the matrices are not equal. This process is crucial as it forms the simplest condition required for matrix equality, making it a foundational aspect of linear algebra.

Solving Equations

Once you have set the condition that corresponding elements of matrices \(A\) and \(B\) must be equal, the next step involves resolving these conditions as equations. For example, if elements \(A_{11}\) and \(B_{11}\) are defined as 3 and \(b\), respectively, the equation \(3 = b\) naturally arises from the equality condition.
The task becomes straightforward: find the values of the variables that satisfy these equations. To illustrate:

• From \(3 = b\), we simply deduce that \(b = 3\).
• For \(a = -5\), it is already evidently provided from the equality condition \( A_{22} = B_{22} \).

Solving these equations helps complete the picture, ensuring the matrices are indeed equal. These simplified equations not only make calculations easier but also allow for easy verification by substituting the values back into the matrix.

Matrix Elements

The building blocks of matrices are their elements, each serving an essential role in the structure and function of a matrix. The term "element" refers to each individual number contained in a matrix. Elements are typically denoted by two indices: the first indicates the row number, and the second denotes the column number in which the element is located.
For example, in the matrix \(A\) given above:

• Element \(A_{12}\) denotes the element in the first row and second column, which is 4.
• Similarly, \(A_{21}\) would be -1.

Identifying and working with matrix elements is fundamental to performing operations such as matrix addition, multiplication, and transposition. The ability to pinpoint and manipulate these individual components is crucial when solving problems involving matrices, including establishing matrix equality.