Problem 5 - Algebra and Trigonometry

Question

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A .\) $$ A=\left[\begin{array}{rrr}{1} & {3} & {-1} \\ {1} & {4} & {0} \\ {-1} & {-3} & {2}\end{array}\right] \quad B=\left[\begin{array}{rrr}{8} & {-3} & {4} \\\ {-2} & {1} & {-1} \\ {1} & {0} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

Both AB and BA result in identity matrices; hence B is the inverse of A.

Step 1: Calculate Product AB

To multiply matrices, use the rule: The element in row i and column j of the product is the dot product of the ith row of the first matrix and the jth column of the second matrix. Let's compute each element of the matrix product AB: 1. First row first column: \(1 \cdot 8 + 3 \cdot (-2) + (-1) \cdot 1 = 8 - 6 - 1 = 1\)2. First row second column: \(1 \cdot (-3) + 3 \cdot 1 + (-1) \cdot 0 = -3 + 3 + 0 = 0\)3. First row third column: \(1 \cdot 4 + 3 \cdot (-1) + (-1) \cdot 1 = 4 - 3 - 1 = 0\)4. Repeat the computation for the remaining elements of AB to get: \[AB = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]. This is the identity matrix.

Step 2: Calculate Product BA

Next, let's compute the product of B and A. Notice each computation mirrors those in the previous step but now using matrices B and A:1. First row first column: \(8 \cdot 1 + (-3) \cdot 1 + 4 \cdot (-1) = 8 - 3 - 4 = 1\)2. First row second column: \(8 \cdot 3 + (-3) \cdot 4 + 4 \cdot (-3) = 24 - 12 - 12 = 0\)3. First row third column: \(8 \cdot (-1) + (-3) \cdot 0 + 4 \cdot 2 = -8 + 0 + 8 = 0\)4. Follow the same process for all elements to obtain: \[BA = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]. This is also an identity matrix.

Key Concepts

Matrix Multiplication

Matrix multiplication is a critical operation in linear algebra that combines two matrices to form a new matrix. It involves calculating the sum of the products of corresponding elements from the rows of the first matrix and the columns of the second matrix. Here are the basic steps to understand this concept:

• For two matrices, say \(A\) and \(B\), the number of columns in \(A\) must match the number of rows in \(B\) for the multiplication to be valid.
• The element in the resulting matrix at position \((i, j)\) is given by the dot product of the \(i\)th row of \(A\) and the \(j\)th column of \(B\).
• Through dot product calculation, the element is added together to givethe final value in the \((i, j)\) position.

Understanding these steps is key when verifying if one matrix is the inverse of another. In the exercise provided, we verify this by showing that both \(A \cdot B\) and \(B \cdot A\) result in the identity matrix.

Identity Matrix

An identity matrix is an important concept in linear algebra, particularly when discussing inverses. It acts as the multiplicative identity in matrix operations, much like the number "1" in real number multiplication. Key features include:

• For an "\(n \times n\)" matrix, the identity matrix, usually denoted as \(I_n\), has ones on its main diagonal (from top left to bottom right) and zeroes elsewhere.
• Multiplying any matrix by an identity matrix returns the original matrix, i.e., \(AI = IA = A\).
• In the context of inverses, if a matrix \(B\) satisfies \(AB = I\) and \(BA = I\), we call \(B\) the inverse of \(A\).

This identity property is a powerful tool used to confirm that one matrix is indeed the inverse of another. In the exercise, we computed \(AB\) and \(BA\) both to be identity matrices, indicating successfully that \(B\) is indeed the inverse of \(A\).

Linear Algebra

Linear algebra is the branch of mathematics focusing on vector spaces and linear mappings between these spaces. It forms the foundation for understanding many operations like matrix multiplication and finding an inverse. Some core ideas include:

• Vector spaces: Collections of vectors where vectors can be added together and multiplied by scalars.
• Matrices: Rectangular arrays of numbers that can represent linear transformations or systems of linear equations.
• Matrix operations: Include addition, scalar multiplication, and more complex operations like matrix multiplication as seen in the exercise.
• Inverses: A matrix may have an inverse, which undoes its transformation effects. Verifying inverses involves checking if products of matrices result in an identity matrix.

Understanding these fundamental concepts provides a solid ground to tackle exercises involving matrix operations, and validate properties like inverses effectively. In the given problem, applying these concepts helped us confirm that matrix \(B\) was indeed the inverse of matrix \(A\) by demonstrating that both \(A \cdot B\) and \(B \cdot A\) equaled the identity matrix.