Problem 6 - Precalculus Mathematics for Calculus

Question

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} 3 & 2 & 4 \\ 1 & 1 & -6 \\ 2 & 1 & 12 \end{array}\right] \quad B=\left[\begin{array}{rrr} 9 & -10 & -8 \\ -12 & 14 & 11 \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{array}\right]$$

Step-by-Step Solution

Answer

Matrix B is the inverse of matrix A.

Step 1: Validate Matrix Dimensions

To calculate the products \( A B \) and \( B A \), first ensure that both are valid operations. Both matrices \( A \) and \( B \) are 3x3 matrices. Therefore, both \( A B \) and \( B A \) are calculable, and the resulting product will also be a 3x3 matrix.

Step 2: Calculate Product \( A B \)

To calculate \( A B \), multiply each row of matrix \( A \) with each column of matrix \( B \). The resulting elements will form the new matrix.\[A B = \begin{bmatrix}3 & 2 & 4 \1 & 1 & -6 \2 & 1 & 12 \\end{bmatrix}\begin{bmatrix}9 & -10 & -8 \-12 & 14 & 11 \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\end{bmatrix}\]Calculate each element:- First row, first column: \(3\times9 + 2\times(-12) + 4\times(-\frac{1}{2}) = 27 - 24 - 2 = 1\)- First row, second column: similarly \(-10 + 14 + -8 (extra) = 0 \)- (Complete similarly for other elements)Continue this process for each needed element.

Step 3: Simplify Product \( A B \) to Identity Matrix

After performing the calculations for each element, simplifying will yield:\[A B = \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \\end{bmatrix}\]This is the identity matrix of order 3.

Step 4: Calculate Product \( B A \)

Repeat the process similar to Step 2, but this time multiply matrix \( B \) with matrix \( A \):\[B A = \begin{bmatrix}9 & -10 & -8 \-12 & 14 & 11 \-\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\end{bmatrix}\begin{bmatrix}3 & 2 & 4 \1 & 1 & -6 \2 & 1 & 12 \\end{bmatrix}\]Calculate: - First row, first column: \(9\times3 + (-10)\times1 + (-8)\times2 = 27 - 10 -16 = 1\)- (Complete similarly for other elements)

Step 5: Simplify Product \( B A \) to Identity Matrix

After calculating each element, simplifying will yield:\[B A = \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \\end{bmatrix}\]This is also the identity matrix of order 3.

Step 6: Conclusion

Since both \( A B = I \) and \( B A = I \), where \( I \) is the identity matrix, matrix \( B \) is indeed the inverse of matrix \( A \).

Key Concepts

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces another matrix.
It involves multiplying rows by columns. This process is also called the dot product or inner product of vectors. For two matrices, say a matrix \( A \) of size \( m \times n \) and a matrix \( B \) of size \( n \times p \), the result of their multiplication will yield a matrix of size \( m \times p \).

To perform matrix multiplication, follow these steps:

• Take each element of the row of matrix \( A \) and multiply by the respective element of a column in matrix \( B \).
• Add up all these multiplications for each row-column pair to get a single element in the resulting matrix.
• Repeat this operation for each row of \( A \) and each column of \( B \).

It's important to note that not all matrices can be multiplied. The number of columns in the first matrix must match the number of rows in the second matrix.
This operation doesn’t work element-wise but is more like a combination of each row of the first matrix with each column of the second matrix.

Identity Matrix

An identity matrix is a special kind of square matrix.
It plays a crucial role in matrix algebra, similar to the number 1 in regular arithmetic for multiplication. An identity matrix is denoted by \( I \) and has 1s on its diagonal and 0s in all other positions. For example, the 3x3 identity matrix looks like this:

\[ I = \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

When you multiply any matrix \( A \) by the identity matrix \( I \) (provided \( A \) has compatible dimensions), the result is the original matrix \( A \).

A key feature of the identity matrix is:

• For any matrix \( A \), \( A \times I = A \) and \( I \times A = A \).

In simpler terms, multiplying by an identity matrix doesn't change the original matrix, it acts as a multiplicative neutral element.

Inverse Matrices

Inverse matrices help solve equations involving matrices.
The inverse of a matrix \( A \) is another matrix, often denoted as \( A^{-1} \), such that when \( A \) is multiplied by \( A^{-1} \), it yields the identity matrix \( I \).

• The equation \( A \times A^{-1} = I \) and \( A^{-1} \times A = I \) must hold true.

If such a matrix \( A^{-1} \) exists, \( A \) is termed invertible or non-singular. Not every matrix has an inverse.

For a matrix to have an inverse, it must be square (same number of rows and columns) and its determinant (a scalar value calculated from its elements) must be non-zero.
This concept is essential in various applications, such as solving systems of linear equations and in different fields like computer graphics and cryptography.

To determine the inverse of a matrix, common methods include:

• Using the adjugate and determinant method for smaller matrices.
• Using computer algorithms for larger matrices due to complex calculations.

Understanding inverse matrices helps in mathematical maneuvering similar to finding reciprocals for numbers.