Problem 24 - 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 \(A^{-1} A=I\) $$ \text { 24. } A=\left[\begin{array}{rrr} {2} & {4} & {-4} \\ {1} & {3} & {-4} \\ {2} & {4} & {-3} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of a provided matrix is obtained after performing relevant row operations on an augmented matrix [A | I] so as to achieve the [I | B] form. After the extraction of matrix B, it is verified that B is the inverse of A by confirming that \(AB = I\) and \(BA = I\).

Step 1: Form Augmented Matrix

Create an augmented matrix by combining the given matrix \(A\) and the identity matrix \(I\) of same order.

Step 2: Transform into Row Echelon Form

Perform row operations so that the left side of the augmented matrix resembles an identity matrix. Typically, the row operations are: Swap two rows, Multiply a row by a non-zero scalar, Add a multiple of one row to another row. Keep doing these operations until the left side of the augmented matrix transforms into an identity matrix.

Step 3: Extract Inverse Matrix

After successfully obtaining the [I | B] form, extract the matrix \(B\), i.e., the right side of the augmented matrix which represents the inverse of the given matrix \(A\).

Step 4: Verification

Multiply the given matrix \(A\) with the extracted inverse \(B = A^{-1}\) from both sides and ensure that you get the identity matrix in both cases, i.e., \(AB = I\) and \(BA = I\).

Key Concepts

Row Operations

Row operations are fundamental techniques used in linear algebra to manipulate matrices. They play a critical role when attempting to invert a matrix. In matrix inversion, row operations help transform the given matrix into a form where it can easily reveal its inverse.

• Swapping Rows: You may interchange any two rows to move elements into more advantageous positions, often to get zeros below a pivot for upper triangular form.


• Row Multiplication: A row can be multiplied by a non-zero scalar, which can simplify matrix elements and help establish leading ones in a row.


• Adding Rows: By adding or subtracting multiples of rows, you can create zeros in specific positions, helping to reach the desired echelon form.

Understanding these operations is crucial because they maintain the underlying properties of the matrix while reshaping it into a form that reveals its inverse.

Augmented Matrix

An augmented matrix combines two matrices into one, which is extremely useful in solving systems of linear equations and finding matrix inverses. In the context of matrix inversion, this involves combining the original matrix, denoted as \(A\), with the identity matrix of the same size.

• The matrix \(A\) represents the coefficients from a system of equations, or the matrix you wish to invert.


• The identity matrix, \(I\), represents the baseline for the solution, as it helps extract the inverse on completion.

By carefully manipulating this augmented form using row operations, you can systematically convert the \([A | I]\) augmented matrix to the impressive \([I | B]\) form, where \(B\) is the sought-after inverse of \(A\).
The process relies on converting the left block of the augmented matrix into the identity matrix while transforming the right block into the inverse.

Identity Matrix

The identity matrix is a special kind of square matrix that plays a pivotal role in linear algebra, especially in matrix inversion. It acts as the "one" in matrix multiplication, similar to how the number one acts in numerical multiplication.

• The identity matrix, typically denoted as \(I\), has ones on the diagonal and zeros elsewhere. For a matrix of size \(n\times n\), it is represented as: \[ I = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix} \]


• Its crucial property is that when any matrix \(A\) of the same dimensions is multiplied by \(I\), it remains unchanged: \(AI = A\) and \(IA = A\).

Inverting a matrix essentially involves transforming it into an identity matrix using the augmented matrix \([A | I]\) and applying row operations. The identity matrix's structure allows the isolation of the inverse in the augmented system, demonstrating how \(A^{-1}\) reverses the effects of \(A\) itself.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra and plays a key role in verifying the correctness of an inverse matrix. It involves combining two matrices to produce a new matrix by systematically computing the dot product of rows from the first with columns from the second.

• For matrices \(A\) and \(B\), the product \(AB\) results in a new matrix, contingent on the compatibility of dimensions (e.g., if \(A\) is \(m\times n\), then \(B\) should be \(n\times p\)).


• The element in the \(i^{th}\) row and \(j^{th}\) column of the resultant matrix is calculated by taking the sum of the products of corresponding elements: \( (AB)_{ij} = \sum_{k} A_{ik}B_{kj} \).

This operation is pivotal in verifying whether a matrix inversion is correct. After identifying \(B = A^{-1}\), if \(AB = I\) and \(BA = I\), where \(I\) is the identity matrix, the inverse computation is confirmed correct.
This operation checks and balances the accuracy of inversions, grounding them in principles of matrix algebra.