Problem 8 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Matrix B is the multiplicative inverse of A if both the products AB and BA yield the identity matrix. If this is not the case, then B is not the inverse of A.

Step 1: Compute the Product AB

We can begin by multiplying matrix A and B. This is done by performing a row by column multiplication, as per the formula for matrix multiplication. Remember, when calculating the elements of the resulting matrix, sum up the outcomes of multiplying the corresponding elements of a row of the first matrix with a column of the second matrix.

Step 2: Compare the Product to the Identity Matrix

After finding the product AB, compare it with the identity matrix. If the product is the same as the identity matrix, this shows that B is the multiplicative inverse of A.

Step 3: Compute the Product BA

Repeat the process of matrix multiplication, but this time for the product of B times A. Again, check if the result is identical to the identity matrix. If it matches, this further confirms that B is the multiplicative inverse of A.

Key Concepts

Matrix Multiplication

Matrix multiplication is a cornerstone operation in linear algebra, particularly essential when working with transformations and systems of equations. Unlike scalar multiplication, which involves simply multiplying each element of a matrix by a constant, matrix multiplication involves a process commonly referred to as 'row by column multiplication'.

Indeed, when we multiply two matrices, the entry in the resulting matrix at position (i, j) is found by taking the dot product of the ith row of the first matrix with the jth column of the second matrix. This process must be repeated for each entry in the resulting matrix. It’s crucial to note that matrix multiplication is not commutative; the order of multiplication matters. Thus, the product of matrix A times B (\( AB \)) can be vastly different from the product of B times A (\( BA \)).

Moreover, the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be defined. If matrix A is of size m x n and matrix B is n x p, their product AB will be a matrix of size m x p. This dimension compatibility is fundamental to performing matrix multiplication correctly.

Identity Matrix

The identity matrix plays a pivotal role in matrix algebra, acting similarly to the number 1 in scalar multiplication. It is a special square matrix that, when multiplied by any compatible matrix, leaves the original matrix unchanged. Represented as \( I \) with varying sizes, an identity matrix is always square and features 1s on the main diagonal (from top left to bottom right) with 0s filling up the rest of the matrix.

In the context of the inverse of a matrix, the identity matrix serves as a litmus test. If matrix B is indeed the multiplicative inverse of matrix A, then both products \( AB \) and \( BA \) must result in the identity matrix corresponding to the size of A and B. If either product does not yield the identity matrix, B cannot be considered the inverse of A.

This property is essential in various mathematical operations and is often a goal when finding the inverse of a given matrix—a crucial step for solving linear equations and for matrix division.

Row by Column Multiplication

Row by column multiplication is the specific method employed in matrix multiplication. This technique involves taking a single row from the first matrix and a single column from the second matrix and multiplying them elementwise, adding up the products to get a single entry in the resultant matrix.

Example:

To multiply a 2x3 matrix, A, by a 3x2 matrix, B, and find the entry in the first row and first column of the resulting matrix, we would perform the following calculation:

• Take the first row of A: \(a_{11}, a_{12}, a_{13}\)
• Take the first column of B: \(b_{11}, b_{21}, b_{31}\)
• Multiply each corresponding element: \(a_{11}*b_{11} + a_{12}*b_{21} + a_{13}*b_{31}\)
• Add the products to find the resulting entry.

The final step would need to be repeated for each entry in the resulting product matrix. This method ensures that each new entry is a summary of the interactions between the corresponding rows and columns of the original matrices and forms one of the fundamental operations for understanding and applying matrix algebra.