Question

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrr}-2 & 2 & 3 \\ 1 & -1 & 0 \\ 0 & 1 & 4\end{array}\right], B=\frac{1}{3}\left[\begin{array}{rrr}-4 & -5 & 3 \\ -4 & -8 & 3 \\ 1 & 2 & 0\end{array}\right]\)

Short answer

After performing the matrix multiplication, it can be seen that both \(AB\) and \(BA\) result in the Identity matrix. Therefore, \(B\) is indeed the inverse of \(A\).

Step-by-step solution

Multiply Matrix \(A\) by \(B\)

Firstly, perform the matrix multiplication of \(A\) and \(B\). The element in the \(i\)-th row and \(j\)-th column of the resulting matrix is calculated as the dot product of the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\).

Cross check by Multiplying Matrix \(B\) by \(A\)

Now, perform the matrix multiplication of \(B\) and \(A\), again with the dot product method. This gives us another matrix.

Check if both matrices are Identity matrices

Lastly, verify if both resulting matrices from step 1 and step 2 are Identity matrices. An Identity Matrix is a square matrix where all elements of the principal (top-left to bottom-right) diagonal are ones and all other elements are zeros.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra important for understanding how matrices interact. The process involves taking two matrices, such as our matrices \( A \) and \( B \), and creating a new matrix based on the sum of products of their rows and columns. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
In our exercise, matrix \( A \) has 3 columns and matrix \( B \) has 3 rows, making them compatible for multiplication. The dot product is computed by multiplying each element of the row of the first matrix by the corresponding element of the column of the second matrix and then summing these products to get each element for the product matrix.
This process is repeated for each row in the first matrix and each column in the second matrix, leading to a new matrix of potentially new dimensions with a well-defined layout of elements. Understanding this concept is crucial for solving more complex mathematical problems in matrix algebra.

Identity Matrix

The identity matrix is a special type of square matrix that plays a role similar to the number '1' in multiplication for real numbers. It is an integral part of matrix algebra and particularly important when dealing with inverse matrices.
An identity matrix is composed of 1s down its principal diagonal (from the top left to the bottom right) and 0s in all other positions. For example, a 3x3 identity matrix would look like this:

• 1 0 0
• 0 1 0
• 0 0 1

When a matrix is multiplied by an identity matrix, it remains unchanged, just like multiplying a number by one. This property makes identity matrices unique in verifying whether a matrix is the inverse of another.
In our exercise, we multiply matrix \( A \) by matrix \( B \) and expect the result to be the identity matrix if \( B \) is indeed the inverse of \( A \). The same result should come when we multiply \( B \) by \( A \). This concept ensures that the operations hold true and the verification is precise.

Matrix Algebra

Matrix algebra is a sophisticated branch of mathematics that uses matrices to solve linear equations and more intricate mathematical problems. It encompasses operations like addition, subtraction, and multiplication of matrices, finding determinants, and understanding inverses and transpositions.
In matrix algebra, inverse matrices are particularly useful. An inverse of a matrix \( A \) is another matrix \( B \), such that when \( A \) is multiplied by \( B \), it results in the identity matrix. This property is crucial for solving systems of linear equations and matrix-related computations successfully.
The exercise illustrates this by determining if \( B \) is the inverse of \( A \). If \( A \cdot B = I \), where \( I \) is the identity matrix, it confirms the inverse relationship. This highlights the power and utility of matrix algebra in providing solutions to mathematical problems that involve linear transformations and systems.
By mastering these elements of matrix algebra, students are better equipped to tackle a wide range of mathematical challenges.