Problem 6 - Algebra and Trigonometry

Question

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{ll} {4} & {5} \\ {2} & {3} \end{array}\right], \quad B=\left[\begin{array}{rr} {\frac{1}{2}} & {-\frac{5}{2}} \\ {-1} & {2} \end{array}\right] $$

Step-by-Step Solution

Answer

No, B is not the multiplicative inverse of A.

Step 1: Multiply AB

Matrix AB will be obtained by multiplying matrix A with matrix B. \[ AB = \left[\begin{array}{ll} {4} & {5} \end{array}\right] \times \left[\begin{array}{rr} {\frac{1}{2}} & {-\frac{5}{2}} \ {-1} & {2} \end{array}\right] = \left[\begin{array}{ll} {(4*\frac{1}{2} + 5*-1)} & {(4*-\frac{5}{2} + 5*2)} \ {(2*\frac{1}{2} + 3*-1)} & {(2*-\frac{5}{2} + 3*2)}\end{array}\right] = \left[\begin{array}{ll} 0 & 0 \ 0 & 1 \end{array}\right] \]

Step 2: Multiply BA

Matrix BA will be obtained by multiplying matrix B with matrix A. \[ BA = \left[\begin{array}{rr} {\frac{1}{2}} & {-\frac{5}{2}} \ {-1} & {2} \end{array}\right] \times \left[\begin{array}{ll} {4} & {5} \ {2} & {3} \end{array}\right] = \left[\begin{array}{ll} {(\frac{1}{2}*4 + -\frac{5}{2}*2)} & {(\frac{1}{2}*5 + -\frac{5}{2}*3)} \ {(-1*4 + 2*2)} & {(-1*5 + 2*3)} \end{array}\right] = \left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right] \]

Step 3: Check if B is the multiplicative inverse of A

Evident from the products AB and BA, neither of the product matrices is an identity matrix. It is thus clear that B is not the multiplicative inverse of A.

Key Concepts

Multiplicative Inverse

When talking about matrices, the concept of a multiplicative inverse is similar to the idea of finding the reciprocal of a number. If you have a matrix \( A \), its multiplicative inverse \( B \) can be defined such that when you multiply \( A \) by \( B \), the result is the identity matrix, say \( I \). Likewise, if you multiply \( B \) by \( A \), you should also get the identity matrix again.

Here's what the formula looks like mathematically:

• \( AB = I \)
• \( BA = I \)

Finding a multiplicative inverse for a matrix isn't always possible. A matrix must be square (having the same number of rows as columns) and its determinant must not be zero for an inverse to exist.

In our exercise, when you multiply matrix \( A \) by matrix \( B \) and vice versa, neither product equals the identity matrix. Therefore, matrix \( B \) is not the multiplicative inverse of matrix \( A \). Remember that checking both products \( AB \) and \( BA \) is crucial.

Identity Matrix

The identity matrix is a special type of square matrix that plays a key role in matrix algebra. It is essentially the matrix equivalent of the number 1, which is the neutral element in multiplication for real numbers.

Key features of an identity matrix include:

• It is always square (e.g., 2x2, 3x3).
• The diagonal elements are 1s.
• All other elements are 0s.

Mathematically, for a 2x2 matrix, it's represented as:\[I = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]
If you multiply any matrix by the identity matrix, the result is the original matrix itself. This property is fundamental when dealing with multiplicative inverses, as seen in the exercise above. The identity matrix allows us to determine whether \( B \) is indeed the inverse of \( A \), which in this case, it is not.

Matrix Algebra

Matrix algebra involves various operations like addition, subtraction, and multiplication of matrices. One of the most essential operations is matrix multiplication, which goes beyond simply multiplying corresponding elements.

Some important points about matrix multiplication include:

• The number of columns in the first matrix must match the number of rows in the second matrix.
• Matrix multiplication is not commutative, meaning \( AB \) does not necessarily equal \( BA \).
• The resulting matrix has dimensions that are determined by the rows of the first matrix and the columns of the second matrix.

When working with matrices, understanding these basics helps solve complex problems such as finding multiplicative inverses and verifying the properties of various matrices. Effective use of matrix algebra techniques can significantly simplify many real-world computations involving data, transformations, and more.