Problem 4 - College Algebra

Question

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

Step-by-Step Solution

Answer

Matrix B is not the multiplicative inverse of Matrix A

Step 1: Compute product A*B

To solve \(AB\), each element of the product is determined by the dot product of the respective row from matrix A and column from matrix B. \[ AB = \left[\begin{array}{rr} -2*1 + 4*-1 & -2*2 + 4*-2 \ 1*1 + -2*-1 & 1*2 + -2*-2 \end{array}\right] = \left[\begin{array}{rr} -6 & -12 \ 3 & 6 \end{array}\right]\]

Step 2: Compute product B*A

Similarly, to solve \(BA\), each element of the product is determined by the dot product of the respective row from matrix B and column from matrix A. \[ BA = \left[\begin{array}{rr} 1*-2 + 2*1 & 1*4 + 2*-2 \ -1*-2 + -2*1 & -1*4 + -2*-2 \end{array}\right] = \left[\begin{array}{rr} 0 & 0 \ 0 & 0 \end{array}\right] \]

Step 3: Compare the products with the identity matrix

The identity matrix is a 2x2 matrix with ones on the diagonal and zeros elsewhere, represented as: \[ I = \left[\begin{array}{rr} 1 & 0 \ 0 & 1 \end{array}\right]\]. However, neither \(AB\) nor \(BA\) equals the identity matrix, so matrix \(B\) is not the multiplicative inverse of matrix \(A\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It's quite different from regular multiplication that we use with numbers. When multiplying two matrices, say matrix \(A\) of size \(m \times n\), and matrix \(B\) of size \(n \times p\), the result is a new matrix \(C\) of size \(m \times p\). This only works when the number of columns in the first matrix (\(A\)) is equal to the number of rows in the second matrix (\(B\)).
In the original exercise, when calculating \(AB\), we multiply each row in matrix \(A\) by each column in matrix \(B\). This process involves taking the dot product of the corresponding elements and summing them up.

• Matrix \(A\) has elements of size 2x2.
• Matrix \(B\) is also 2x2.
• The resulting matrix from multiplying \(A\) and \(B\) will have dimensions 2x2.

This operation is useful when dealing with systems of equations, transformations, and many applications across physics and computer graphics.

Dot Product

The dot product is a fundamental operation that combines vectors to produce a scalar and is a building block for matrix multiplication. When performing matrix multiplication, each element of the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.
To compute a dot product, multiply corresponding elements from the row vector and column vector, then sum these products. For example, to find the element in the first row and first column of product \(AB\), multiply the corresponding entries from \(A\) and \(B\):\(-2 \times 1 + 4 \times -1 = -6\).

• A helps establish the outcome of transformations or rotations in matrices.
• The dot product is also used to compute angles between vectors and analyze the orthogonality.

This operation is easy but crucial to understand, especially when it comes to verifying matrix inverses or assessing matrix dynamics.

Identity Matrix

The identity matrix plays a pivotal role in matrix algebra, resembling the number 1 in regular arithmetic. When any matrix is multiplied by an identity matrix, it retains its original form, serving as a neutral element in matrix multiplication.
For a 2x2 identity matrix, it is represented as \[I = \left[\begin{array}{rr} 1 & 0 \ 0 & 1 \end{array}\right]\]The importance of the identity matrix is seen in solving systems of linear equations and finding inverses. If you multiplied a matrix with its inverse and got the identity matrix, you confirmed that they are indeed inverses of each other.

• In essence, the identity matrix allows us to perform safe transformations without changing the original properties of the matrix.
• It's as if we are multiplying by 1 or adding 0 when it comes to matrix operations.

In our original problem, neither product \(AB\) nor \(BA\) resulted in an identity matrix, which tells us that \(B\) is not the inverse of \(A\).

Matrix Algebra

Matrix algebra is the set of rules and operations for matrices, and it extends many of the concepts from simple algebra but into higher dimensions. It includes operations like addition, subtraction, and multiplication, as well as concepts like inverses and transposes.
Understanding the properties of matrices, such as commutativity (where \(A \times B\) is not necessarily equal to \(B \times A\)), is crucial. In matrix algebra, finding an inverse is key, especially when solving systems of equations. An inverse matrix, when it exists, is like a "reverse operation," and multiplying a matrix by its inverse yields the identity matrix.

• Significance in linear transformations and space projections.
• Crucial for mathematical models in engineering and computer science.

In our scenario, since neither \(AB\) nor \(BA\) equaled the identity matrix, it shows how the theoretical beauty of matrix algebra helps us quickly assess relationships and operations between matrices.