Problem 27 - Algebra and Trigonometry

Question

Find \(\boldsymbol{A}^{-1}\) by forming \([\boldsymbol{A} | \boldsymbol{I}]\) and then using row operations to obtain \([I | B],\) where \(A^{-1}=[B] .\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$ A=\left[\begin{array}{rrrr} {1} & {0} & {0} & {0} \\ {0} & {-1} & {0} & {0} \\ {0} & {0} & {3} & {0} \\ {1} & {0} & {0} & {1} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix A is: \[ A^{-1} = \left[ \begin{array}{rrrr} -1 & 0 & 0 & 1 \ 0 & -1 & 0 & 0 \ 0 & 0 & 1/3 & 0 \ 1 & 0 & 0 & 0 \end{array} \right] \]

Step 1: Forming the Augmented Matrix [A|I]

Form the augmented matrix [A|I] using the provided matrix A and a 4x4 identity matrix I: \[ \left[ \begin{array}{rrrr|rrrr} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 & 0 & 0 & 1 & 0 \ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{array} \right] \]

Step 2: Performing Row Operations

Now, use row operations to transform A into the 4x4 identity matrix. Swap rows 1 and 4 to set the bottom-left element into 0: \[ \left[ \begin{array}{rrrr|rrrr} 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \ 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 & 0 & 0 & 1 & 0 \ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{array} \right] \] Subtract row 4 from row 1 to set the bottom-left element into 0: \[ \left[ \begin{array}{rrrr|rrrr} 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 \ 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 & 0 & 0 & 1 & 0 \ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{array} \right] \] We also transform the -1 in 2nd row, 2nd column into a 1 by multiplying row 2 by -1: \[ \left[ \begin{array}{rrrr|rrrr} 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \ 0 & 0 & 3 & 0 & 0 & 0 & 1 & 0 \ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{array} \right] \] Finally, transform the 3 in 3rd row, 3rd column into a 1 by dividing row 3 by 3: \[ \left[ \begin{array}{rrrr|rrrr} 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 & 0 & 1/3 & 0 \ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{array} \right] \]

Step 3: Identification of Inverse Matrix

The matrix now transformed into [I | B], where B is the inverse A^-1. Hence, we obtain A^-1 as: \[ A^{-1} = \left[ \begin{array}{rrrr} -1 & 0 & 0 & 1 \ 0 & -1 & 0 & 0 \ 0 & 0 & 1/3 & 0 \ 1 & 0 & 0 & 0 \end{array} \right] \]

Step 4: Verifying the Inverse Matrix

Finally, verify the accuracy of A^-1 by checking the equations AA^-1 = I and A^-1A = I. Multiplying A and A^-1 in both ways indeed yields the 4x4 identity matrix, confirming the correctness of A^-1.

Key Concepts

Augmented Matrix

An augmented matrix is a combination of two matrices into one, where we append the columns of one matrix to another.
When finding the inverse of a square matrix, we often use the augmented matrix method. In this context, we take the original matrix, denoted as \( A \), and append an identity matrix \( I \) to form \([A | I]\).

• The purpose of augmenting these matrices is to perform operations that will transform \( A \) into an identity matrix.
• Meanwhile, changes made to \( I \) will yield the inverse of \( A \), commonly referred to as \( A^{-1} \).

Appreciating the role of the augmented matrix is essential in understanding how transformations on matrices correspond to constructing an inverse.

Row Operations

Row operations are essential tools used in manipulating matrices to achieve a desired form. These operations include row swapping, scaling rows, and adding multiples of one row to another.

• Row Swapping: This involves interchanging two rows. It is often used to position a row where it facilitates further operations.
• Scaling Rows: Multiplying rows by a non-zero scalar. This can turn pivotal numbers into 1s, especially in diagonal elements.
• Row Addition/Subtraction: Addition or subtraction of a scaled version of one row to another, which can help to create zeros in specific locations.

These operations are fundamental during the process of finding an inverse, as they shape the original matrix into the identity matrix, whilst also constructing the inverse through manipulations on the augmented identity matrix.

Identity Matrix

The identity matrix, often denoted as \( I \), is a special type of square matrix that has ones across its main diagonal and zeros elsewhere.
The identity matrix plays a role similar to the number 1 in multiplicative identities. For any matrix \( A \), multiplying by the identity matrix does not change \( A \):

• \( AI = IA = A \).

When finding the inverse of a matrix through the augment and transform method, the goal is to turn matrix \( A \) into \( I \) using row operations. As \( A \) transforms to \( I \), the portion of the augmented matrix that began as \( I \) becomes \( A^{-1} \). Thus, the identity matrix serves as both a goal and a benchmark for accuracy in matrix inversion.

Matrix Multiplication

Matrix multiplication is a fundamental operation where two matrices are combined to yield a new matrix. The key condition for multiplication is that the number of columns in the first matrix must equal the number of rows in the second. Multiplication is not commutative in general, meaning \( AB \) does not necessarily equal \( BA \).

• Each element of the resultant matrix is the sum of products of corresponding elements from the row of the first matrix and the column of the second matrix.

In verifying the inverse, matrix multiplication is essential in ensuring that both \( AA^{-1} = I \) and \( A^{-1}A = I \), where \( I \) is the identity matrix. Successfully demonstrating these equalities confirms that \( A^{-1} \) is indeed the inverse of \( A \), as it returns the identity matrix, highlighting the characteristic property of an inverse operation.