Problem 21 - 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 \(A^{-1} A=I\) $$ A=\left[\begin{array}{rrr} {1} & {2} & {-1} \\ {-2} & {0} & {1} \\ {1} & {-1} & {0} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix \(A^{-1}\) is \[ B = \left[\begin{array}{rrr}0 & -2 & -6\0.5 & -0.5 & -1\0 & -3 & -4\end{array}\right] \]

Step 1: Form the Augmented Matrix

Start by forming the augmented matrix [A | I]. Here, I is the 3x3 identity matrix. \[ A | I = \left[\begin{array}{rrr|rrr}1 & 2 & -1 & 1 & 0 & 0\-2 & 0 & 1 & 0 & 1 & 0\1 & -1 & 0 & 0 & 0 & 1\end{array}\right] \]

Step 2: Row Operation 1

Perform the row operations R2+2R1->R2 and R3-R1->R3 to make all the other elements of the first column zero while keeping as it is. \[ \left[\begin{array}{rrr|rrr}1 & 2 & -1 & 1 & 0 & 0\0 & 4 & -1 & 2 & 1 & 0\0 & -3 & 1 & -1 & 0 & 1\end{array}\right] \]

Step 3: Row Operation 2

Now, we adjust the second row to have a leading 1, by performing the operation R2/4->R2. \[ \left[\begin{array}{rrr|rrr}1 & 2 & -1 & 1 & 0 & 0\0 & 1 & -0.25 & 0.5 & 0.25 & 0\0 & -3 & 1 & -1 & 0 & 1\end{array}\right] \]

Step 4: Row Operation 3

Using the second row as reference, perform row operations R1-2R2->R1 and R3+3R2->R3 to make all the other elements of the second column zero.\[ \left[\begin{array}{rrr|rrr}1 & 0 & -1.5 & 0 & -0.5 & 0\0 & 1 & -0.25 & 0.5 & 0.25 & 0\0 & 0 & -0.25 & 0 & 0.75 & 1\end{array}\right] \]

Step 5: Row Operation 4

Remember, the identity matrix I should be in the left side. Hence, perform the row operation -4R3->R3.\[ \left[\begin{array}{rrr|rrr}1 & 0 & -1.5 & 0 & -0.5 & 0\0 & 1 & -0.25 & 0.5 & 0.25 & 0\0 & 0 & 1 & 0 & -3 & -4\end{array}\right] \]

Step 6: Row Operation 5

Now, perform the final row operation R1 + 1.5R3->R1 and R2 + 0.25R3->R2 to obtain the identity matrix on the left side.\[ [I | B] = \left[\begin{array}{rrr|rrr}1 & 0 & 0 & 0 & -2 & -6\0 & 1 & 0 & 0.5 & -0.5 & -1\0 & 0 & 1 & 0 & -3 & -4\end{array}\right] \]

Step 7: Extract The Inverse

Finally, extract the inverse matrix \(A^{-1} = B\) from the right side of the augmented matrix.\[ A^{-1} = B = \left[\begin{array}{rrr}0 & -2 & -6\0.5 & -0.5 & -1\0 & -3 & -4\end{array}\right] \]

Step 8: Check The Result

Verify the result by multiplying the original matrix with its inverse. The result should be the identity matrix \(I = AA^{-1} = A^{-1}A\). The calculation is lengthy, however, it is essential to check the solution validity.

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool in linear algebra, especially when finding the inverse of a given matrix. When we augment a matrix, we place it next to another matrix with the help of a vertical line to create one larger composite matrix.
In the context of finding an inverse, you augment your given matrix, denoted as \(A\), with the identity matrix, \(I\), leading to \([A | I]\).
This method helps to transform the original matrix into the identity matrix through subsequent row operations. \([A | I]\) setup initially places \(A\) on the left and \(I\) on the right.
This kind of transformation will eventually reveal the inverse of matrix \(A\) on the right side of the transformation. It's a structured method, setting a clear path towards identifying the inverse matrix.

Row Operations

Row operations are used extensively while finding the inverse of a matrix using an augmented matrix. These operations help systematically modify the rows to achieve the desired form, usually the identity matrix on one side of the augmented setup.
The primary types of row operations include:

• Swap: Interchanging two rows
• Scale: Multiplying all elements of a row by a non-zero scalar
• Add/Subtract Multiple: Adding or subtracting a multiple of one row to another

In the given exercise, row operations were applied to \([A|I]\) to step-by-step transform the portion of \(A\) into \(I\).
Each time you cancel, scale, or swap elements, you're guided towards isolating the identity matrix, revealing \(A^{-1}\) on the augmented side. Applying row operations thoughtfully maintains accuracy in the inverse extraction.

Identity Matrix

The identity matrix, often denoted as \(I\), plays a central role when calculating a matrix's inverse. The size of \(I\) corresponds to the size of the square matrix \(A\) that you work with.
An identity matrix is a special case where all the diagonal elements are ones, and all other elements are zeros:

• For a 3x3 matrix, it looks like this: \[I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

In the context of inverse finding, transforming \(A\) into an identity matrix through row operations means that you've successfully determined \(A^{-1}\).
An identity matrix acts as a multiplicative neutral element in matrices, much like multiplying by one does for numbers. Thus, confirming \(A \, A^{-1} = I\) and \(A^{-1} \, A = I\) is crucial for inverse verification.

Inverse Verification

Verifying whether the computed inverse is correct is an essential conclusion to finding the inverse matrix process. The notion of inverse verification involves checking two fundamental equations:
- \(A \, A^{-1} = I\)- \(A^{-1} \, A = I\)
The product of a matrix and its inverse should yield the identity matrix, both when \(A\) post-multiplies \(A^{-1}\) and vice versa.
Multiplying the original matrix with your result for \(A^{-1}\) not only checks your manual row operation accuracy but confirms that the transformation from \([A | I]\) to \[I | B\] was implemented correctly. Engage in detailed matrix multiplication to validate all elements match up with those of the identity matrix.

• This check reassures the coherence of your calculated inverse.
• It also underlines that the algebraic manipulation done leads to meaningful results.