Problem 9 - Algebra for College Students

Question

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{rr} 2 & -1 \\ 0 & -4 \\ -5 & 3 \end{array}\right], \quad B=\left[\begin{array}{rrr} 5 & -2 & 6 \\ -1 & 4 & -2 \end{array}\right] $$

Step-by-Step Solution

Answer

AB exists and is \( \begin{bmatrix} 11 & -8 & 14 \\ 4 & -16 & 8 \\ -28 & 22 & -36 \end{bmatrix} \); BA does not exist.

Step 1: Verify Matrix Dimensions for Multiplication AB

Matrix \(A\) is 3x2 and matrix \(B\) is 2x3. For multiplication \(A \times B\) to be possible, the number of columns in \(A\) must equal the number of rows in \(B\), which they do (both 2). Thus, the product \(AB\) is possible and the resulting matrix will be 3x3.

Step 2: Compute AB Matrix

The matrix product \(AB\) is calculated by taking the dot product of the rows of \(A\) with the columns of \(B\). For each element \(c_{ij}\) in \(C = AB\), compute it as: - \(c_{11} = (2)(5) + (-1)(-1) = 10 + 1 = 11\)- \(c_{12} = (2)(-2) + (-1)(4) = -4 - 4 = -8\)- \(c_{13} = (2)(6) + (-1)(-2) = 12 + 2 = 14\)- \(c_{21} = (0)(5) + (-4)(-1) = 0 + 4 = 4\)- \(c_{22} = (0)(-2) + (-4)(4) = 0 - 16 = -16\)- \(c_{23} = (0)(6) + (-4)(-2) = 0 + 8 = 8\)- \(c_{31} = (-5)(5) + (3)(-1) = -25 - 3 = -28\)- \(c_{32} = (-5)(-2) + (3)(4) = 10 + 12 = 22\)- \(c_{33} = (-5)(6) + (3)(-2) = -30 - 6 = -36\)Thus, \(AB = \begin{bmatrix} 11 & -8 & 14 \ 4 & -16 & 8 \ -28 & 22 & -36 \end{bmatrix}\).

Step 3: Verify Matrix Dimensions for Multiplication BA

Matrix \(B\) is 2x3 and matrix \(A\) is 3x2. For \(BA\) to be possible, the number of columns in \(B\) must equal the number of rows in \(A\). Here, 3 does not equal 3. Thus, the product \(BA\) does not exist.

Key Concepts

Dot Product

The dot product is a fundamental operation used in matrix multiplication. When we multiply two matrices, we actually compute numerous dot products between rows and columns. Let's break this down: you take a row from the first matrix and a column from the second matrix. Then, multiply corresponding elements and add them up. This sum is the dot product.

For example, consider two vectors: \([a_1, a_2]\) and \([b_1, b_2]\). The dot product is calculated as \(a_1 \cdot b_1 + a_2 \cdot b_2\). This same process is used in matrix multiplication, just performed repetitively between matrix rows and columns. This tool allows matrices to combine into new matrices with different dimensions.

In the problem given, when calculating the product \(AB\), every element in the resulting matrix is the dot product of a row from matrix \(A\) and a column from matrix \(B\). Understanding the dot product equips you to handle matrix multiplications effectively.

Matrix Dimensions

Matrix dimensions are foundational for determining if multiplication between two matrices is feasible. The dimensions of a matrix are written as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. When multiplying matrices \(A\) and \(B\), the number of columns in \(A\) must be equal to the number of rows in \(B\).

As a rule of thumb:

• If matrix \(A\) is \(x \times y\)
• Matrix \(B\) must be \(y \times z\)
• The resulting matrix will be \(x \times z\)

This rule is essential in determining whether multiplication is possible and predicting the dimension of the resulting matrix. In our exercise, \(A\) is a \(3 \times 2\) matrix and \(B\) is a \(2 \times 3\) matrix, which makes \(AB\) a \(3 \times 3\) matrix. However, the reverse product \(BA\) is not defined because the dimensions are not compatible (specifically, \(B\)'s columns do not match \(A\)'s rows).

Knowing matrix dimensions enables you to quickly assess the possibility of operations and understand their outcomes.

Matrix Product

The matrix product, crucial in matrix algebra, results from multiplying two matrices. The procedure involves using the dot product across rows and columns. This operation allows us to transform data structures, solve linear equations, and model real-world situations.

Consider the example of multiplying matrices \(A\) and \(B\):

• Matrix \(A\) is \(3 \times 2\)
• Matrix \(B\) is \(2 \times 3\)

Therefore, multiplying \(A\) by \(B\) gives a \(3 \times 3\) matrix. Each element in this resultant matrix is derived from taking the dot product of relevant rows and columns.

This mathematical operation isn't commutative, which means that \(AB eq BA\) in general. In fact, in our solution, we saw \(AB\) exists, but \(BA\) does not due to dimension mismatch. Being aware of what results possible matrix operations yield is vital for further mathematical and technical applications.

Matrix Theory

Matrix theory forms the backbone of linear algebra and provides the basis for various mathematical operations involving matrices. It allows us to explore properties like invertibility, determinants, eigenvalues, and more.

Matrices are more than arrangements of numbers; they represent complex mathematical relationships and transformations. For instance, consider the simplification of linear equations using matrices. You can solve systems of equations more efficiently by representing them in matrix form and then applying various operations and properties from matrix theory.

Matrix theory also highlights transformations such as rotations, scaling, and shear, which are significant in fields like computer graphics and engineering. Understanding the theory not only helps one perform calculations but also provides the foundation to grasp advanced concepts such as linear transformations and vector spaces.

Delving into this subject, you develop problem-solving abilities and better comprehension of how different components interact within mathematical systems.