Problem 10 - Algebra and Trigonometry

Question

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} {0} & {2} & {0} \\ {3} & {3} & {2} \\ {2} & {5} & {1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {-3.5} & {-1} & {2} \\ {0.5} & {0} & {0} \\ {4.5} & {2} & {-3} \end{array}\right] $$

Step-by-Step Solution

Answer

Yes, Matrix \(B\) is the multiplicative inverse of Matrix \(A\).

Step 1: Calculate the product AB

To calculate the product of two matrices, match the first row in the first matrix with the first column in the second matrix. Multiply each pair of elements together and then add these up:\n\n\( (AB)_{ij} = \sum_k a_{ik}*b_{kj} \)\n\nusing which the product AB is calculated to be:\n\n\[AB=\left[\begin{array}{rrr} {1} & {0} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1} \end{array}\right]\]

Step 2: Calculate the product BA

Now, calculate the product BA using the same process as before:\n\n\( (BA)_{ij} = \sum_k b_{ik}*a_{kj} \)\n\nThis gives us the following matrix:\n\n\[BA=\left[\begin{array}{rrr} {1} & {0} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1} \end{array}\right]\]

Step 3: Compare the products AB and BA to an identity matrix

Since both the products AB and BA are the identity matrix, \(B\) is the multiplicative inverse of \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a core operation in linear algebra, essential for various applications in science and engineering. To multiply two matrices, follow these steps:

• Ensure the number of columns in the first matrix matches the number of rows in the second matrix.
• Multiply each element of the row in the first matrix by the corresponding element of the column in the second matrix.
• Add the products to get a single element in the resulting matrix.

Mathematically, this operation is expressed as \((AB)_{ij} = \sum_k a_{ik}b_{kj}\).
This sum iterates over each of the elements aligned from the row of the first matrix and the column of the second matrix. It's crucial to remember matrix multiplication is not commutative, meaning\(AB\) does not necessarily equal \(BA\). It's a calculated process that needs careful indexing and summing of products.

Identity Matrix

The identity matrix is a special type of matrix that acts as the "one" in matrix algebra, similar to how 1 works in regular multiplication. It is an essential concept when discussing matrix inverses.
An identity matrix has 1's on its diagonal and 0's elsewhere:

• In 2x2 form: \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\)
• In 3x3 form: \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\)

When any matrix is multiplied by an identity matrix matching its dimensions, it remains unchanged. This property is crucial for testing multiplicative inverses, as a matrix \(A\) times its inverse \(B\) should yield the identity matrix.

Inverse Matrices

Inverse matrices are defined as matrices that can "reverse" the effect of the original matrix through multiplication. For a matrix \(A\) to have an inverse, it must be square (same number of rows as columns) and non-singular, meaning it has a non-zero determinant.
The purpose of a matrix inverse is that when it's multiplied with the original matrix, it produces the identity matrix. In formula terms, if \(A\) is the matrix and \(B\) is its inverse, then \(AB = BA = I\).

• The existence of an inverse is crucial for solving linear equations represented in matrix form.
• Not every matrix has an inverse; it's only possible under specific conditions relating to linear independence and determinant values.

Determining whether two matrices are inverses involves calculating both \(AB\) and \(BA\) and checking if the resulting product is the identity matrix.

Linear Algebra

Linear algebra is the branch of mathematics focused on vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. It's foundational for many applications in engineering and the physical sciences.
Key elements of linear algebra include:

• Vector Spaces: Collections of vectors where vector addition and scalar multiplication are defined.
• Matrices: Arrays of numbers that represent linear transformations and provide a framework for solving systems of equations.
• Determinants and Eigenvalues: Used to analyze matrix properties and provide insights into linear transformations.

Understanding linear algebra not only involves performing operations with matrices and vectors but also interpreting the geometric and theoretical insights they offer—for example, how transformations affect spaces or the nature of solutions to linear systems.