Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}2 & -17 & 11 \\ -1 & 11 & -7 \\ 0 & 3 & -2\end{array}\right], B=\left[\begin{array}{llr}1 & 1 & 2 \\ 2 & 4 & -3 \\ 3 & 6 & -5\end{array}\right]\)

Step-by-Step Solution

Answer

If both resulting matrices are identity matrices, then \(B\) is the inverse of \(A\). If not, then \(B\) is not the inverse of \(A\).

Step 1: Multiplication of A and B

Multiply the matrices \(A\) and \(B\) together. By multiplying the two matrices, you can do this row by column. For example, the element in the first row and first column of the resulting matrix can be calculated by summing the product of the corresponding elements of the first row of \(A\) and the first column of \(B\). (2*1) + (-17*2) + (11*3) = -29. Continue with this pattern for the remaining elements of the resulting matrix.

Step 2: Multiplication of B and A

Now, multiply the matrices \(B\) and \(A\) together. It's important to remember that matrix multiplication is not commutative - so \(AB\) won't be the same as \(BA\). Repeat the row by column process described in Step 1 to get the second resulting matrix.

Step 3: Check if the Resulting Matrices are Identity Matrices

Now that you have two resulting matrices, check if these are identity matrices. An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation that is central to many mathematical concepts and applications. When multiplying two matrices, such as \(A\) and \(B\), it is essential to understand that the process involves taking the rows of the first matrix and "dotting" them with the columns of the second. This means we perform the following steps:

• For each element of the resulting matrix, calculate by multiplying each element of the row from the first matrix with the corresponding element of the column from the second matrix,
• Then sum these products.

If you have a matrix \(A\) with dimensions \(m \times n\) and a matrix \(B\) with dimensions \(n \times p\), their product \(AB\) will have dimensions \(m \times p\). Note that matrix multiplication is not commutative, meaning that in general, \(AB eq BA\). This characteristic makes it essential to follow the order correctly when solving problems involving matrices.

In our exercise, the task was to multiply matrices \(A\) and \(B\), and then \(B\) and \(A\) to check for special properties.

Identity Matrix

The identity matrix plays the same role in matrix operations as the number 1 does in multiplication of numbers. If you multiply any square matrix by the identity matrix of compatible size, it remains unchanged. In mathematical terms, if \(I\) is the identity matrix, then for any matrix \(M\), \(IM = MI = M\).

An identity matrix is a square matrix where all elements of the principal diagonal are ones and all other elements are zeros. For instance, a 3x3 identity matrix looks like this:
\[I = \begin{bmatrix} 1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{bmatrix}\]
In solving our exercise, checking if the resulting matrices \(AB\) and \(BA\) are identity matrices was crucial because this condition indicates that one matrix is indeed the inverse of the other.

Inverse Matrix

The inverse of a matrix is akin to the reciprocal of a number. If \(A\) is a square matrix, its inverse is denoted \(A^{-1}\), and it must satisfy the condition that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix. This means that when you multiply a matrix by its inverse, you end up with the identity matrix.

Finding the inverse matrix is not always possible; it only exists for square matrices where the determinant is non-zero. In our exercise, we were given matrices \(A\) and \(B\), and the task was to verify that one is the inverse of the other by checking if their product in both orders resulted in the identity matrix.

Thus, if both products \(AB\) and \(BA\) are identity matrices, then \(B\) is indeed the inverse of \(A\), confirming the mathematical rule for inverse matrices.