Problem 38 - Precalculus with Limits

Question

In Exercises 33-40, if possible, find \(AB\) and state the order of the result. \(A=\left[\begin{array}{r} 0 & 0 & 5 \\ 0 & 0 & -3 \\ 0 & 0 & 4 \end{array}\right]\), \(B=\left[\begin{array}{r} 6 & -11 & 4 \\ 8 & 16 & 4 \\\ 0 & 0 & 0 \end{array}\right]\)

Step-by-Step Solution

Answer

The product of matrices \(A\) and \(B\) is a 3x3 matrix with all its elements equal to zero, so \(AB = \left[\begin{array}{r} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{array}\right]\) and the order of the resulting matrix is 3x3.

Step 1: Multiply Matrices

Begin by multiplying the first row of matrix \(A\) by the first column of matrix \(B\), then add the results. Repeat this for each combination of rows from \(A\) and columns from \(B\). Resulting matrix, \(AB\), is \(\left[\begin{array}{r} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{array}\right]\)

Step 2: Determine Order of Result

The order of a matrix is given as 'number of rows x number of columns'. The resulting matrix has 3 rows and 3 columns, so its order is 3x3.

Key Concepts

Matrices

Matrices are fundamental in many areas of mathematics and science. Essentially, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used to solve systems of linear equations, transform geometric shapes, and more.

• Each number in a matrix is called an element.
• Matrices come in various dimensions, like 2x2, 3x3, 3x2, etc.
• The position of elements is defined by their row and column numbers.

Learning how to multiply matrices is a vital skill. This exercise involves multiplying matrices, which requires precision.
Each element of the resultant matrix is obtained by summing up the products of corresponding elements from the rows of the first matrix and the columns of the second. Practice matters, and you'll often find yourself checking whether or not matrix multiplication is possible before starting, as matrices must comply with certain rules regarding their dimensions.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is fundamental to both pure and applied mathematics. Key concepts of linear algebra include:

• Vectors: Essentially a single row or column in a matrix.
• Matrix operations: Including addition, subtraction, and importantly, multiplication.
• Linear transformations: Functions that map vectors to vectors in a linear manner.

In matrix multiplication, which is a core operation in linear algebra, compatibility is paramount. You can only multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix.
This compatibility condition is part of the linear algebra principles essential for successfully engaging in matrix operations. Understanding this ensures that the exercise proceeds with logical and conceptual clarity.

Matrix Order

Matrix order refers to the dimensions of a matrix, expressed as ‘number of rows x number of columns’. It is crucial because it dictates what operations are possible between matrices.

• An m x n matrix has m rows and n columns.
• Matrix order helps to identify valid multiplications or additions.
• In this exercise, the resulting matrix has an order of 3x3, confirming valid multiplication.

The concept of order directly influences the results of matrix multiplication.
You determine the order of the resulting matrix from the multiplication of two matrices by taking the number of rows from the first matrix and the number of columns from the second matrix. Keeping track of dimensions ensures appropriate calculations, contributing to effective problem-solving in matrix-related exercises.