Problem 8 - Precalculus: Mathematics for Calculus

Question

Find the values of \(a\) and \(b\) that make the matrices \(A\) and \(B\) equal. $$A=\left[\begin{array}{rrr} 3 & 5 & 7 \\ -4 & a & 2 \end{array}\right] \quad B=\left[\begin{array}{rrr} 3 & 5 & b \\ -4 & -5 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The values are \(a = -5\) and \(b = 7\).

Step 1: Understand Matrix Equality

For two matrices to be equal, their corresponding elements must be equal. This means that each element in matrix \(A\) must be identical to the element in the same position in matrix \(B\).

Step 2: Compare the First Row Elements

Examine the elements of the first row: \(A[1,3] = 7\) and \(B[1,3] = b\). Since the other elements in the first row are already equal, set \(7 = b\).

Step 3: Solve for b

From the equation \(7 = b\), simply solve to find \(b\). Therefore, \(b = 7\).

Step 4: Compare the Second Row Elements

Look at the second row: \(A[2,2] = a\) and \(B[2,2] = -5\). Set these equal by writing \(a = -5\).

Step 5: Solve for a

From the equation \(a = -5\), we already have the value. Thus, \(a = -5\).

Key Concepts

Matrices

Matrices are a fundamental concept in linear algebra that are used to organize numbers in a rectangular grid, where each number is known as an element.
Matrices can have various dimensions, which are defined by the number of rows and columns they contain. The structure of a matrix allows it to represent complex data and perform calculations like solving equations or transforming geometric figures.
In many applications, you will see matrices used in fields such as physics, engineering, computer science, and economics.

• The elements in a matrix are usually written within square brackets \([ ]\).
• Matrices can be added, subtracted, and multiplied with certain rules.
• They are typically denoted by uppercase letters, such as matrix \(A\) or \(B\), as seen in the original exercise.

Matrix Elements

The elements of a matrix are the individual values organized in rows and columns. It is important to note that each element's position in the matrix is crucial.
For instance, elements are often described by their position in the matrix, like \(A[i,j]\), where \(i\) represents the row number and \(j\) represents the column number.
This notation helps in pinpointing exactly which number you are referring to, especially when visualizing mathematical concepts or solving equations such as making two matrices equal.

• Matrix equality relies heavily on each corresponding element being equal across different matrices.
• In the provided exercise, ensuring \(A[2,2] = a\) is equal to \(B[2,2] = -5\) ensures consistency in position and value.
• Thus, understanding the placement of elements enhances comprehension when comparing matrices.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear transformations, and matrices. It serves as a critical tool for analyzing and solving systems of linear equations.
Understanding matrices is central to this study, as they are foundational to numerous concepts like linear mappings and transformations.
It's widely applied in various disciplines, including computer graphics, machine learning, and data science.

• Linear algebra helps manipulate linear systems—a set of equations where each term is either a constant or the product of a constant and a single variable.
• Matrices simplify the representation of these systems, making it easier to analyze and solve them.
• The equality of matrices can symbolize a particular solution to a system of equations, usually indicating consistent and verified data.