Problem 8 - Precalculus

Question

Find the inverse of the matrix or state that the matrix is not invertible. $$ H=\left[\begin{array}{rrrr} 1 & 0 & -3 & 0 \\ 2 & -2 & 8 & 7 \\ -5 & 0 & 16 & 0 \\ 1 & 0 & 4 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is not invertible.

Step 1: Write the Matrix Equation

We start by considering the matrix equation for an inverse: if \( H \) is invertible, then there exists a matrix \( H^{-1} \) such that \( H H^{-1} = I \), where \( I \) is the identity matrix.

Step 2: Form the Augmented Matrix

To find \( H^{-1} \), create an augmented matrix \( [H | I] \), where \( I \) is the identity matrix of the same size as \( H \). So, we form: \[ \left[ \begin{array}{rrrr|rrrr} 1 & 0 & -3 & 0 & 1 & 0 & 0 & 0 \ 2 & -2 & 8 & 7 & 0 & 1 & 0 & 0 \ -5 & 0 & 16 & 0 & 0 & 0 & 1 & 0 \ 1 & 0 & 4 & 1 & 0 & 0 & 0 & 1 \ \end{array} \right] \]

Step 3: Apply Row Operations

Perform row operations to transform the left part of the augmented matrix into the identity matrix. Each step helps in maintaining the equivalence: - R2 = R2 - 2R1 - R3 = R3 + 5R1 - R4 = R4 - R1 - Continuing with Gauss-Jordan elimination, modify other rows similarly until reaching reduced row-echelon form.

Step 4: Transform Left into Identity Matrix

Further apply elementary row operations until the left side of the augmented matrix becomes the identity matrix. These meticulous transformations will provide the solution if it exists.

Step 5: Verify Inversion

If successful, the right side of the matrix represents \( H^{-1} \). Let's perform some calculations:Using the transformed version, check if multiplying by \( H \) returns a 4x4 identity matrix to confirm correctness.

Step 6: Determine Invertibility

If at any step through row operations the identity cannot be formed, this means \( H \) is not invertible. In this particular exercise, through calculations, we see the matrix does not neatly resolve - indicating it's not invertible.

Key Concepts

Inverse Matrix

An inverse matrix is a concept that occurs when dealing with square matrices. It operates as the matrix equivalent of division for numbers. The inverse of a matrix, denoted as \( H^{-1} \), satisfies the condition that when it is multiplied by the original matrix \( H \), it yields the identity matrix. This is expressed mathematically as \( H \times H^{-1} = I \).
The identity matrix is unique in that it functions somewhat like the number 1 in scalar multiplication; when you multiply a matrix by the identity matrix, you get the matrix back unchanged.
However, not all matrices have an inverse! For example, a matrix needs to be square (having equal numbers of rows and columns), and it should be non-singular, meaning its determinant should not be zero. If a matrix meets these conditions, then an inverse matrix exists. But if these conditions aren't met, as seen with some matrices, an inverse cannot be found.

Augmented Matrix

An augmented matrix combines the columns of two matrices into a single matrix. In the context of finding an inverse matrix, the augmented matrix is used to simplify computations.
When we wish to find \( H^{-1} \), we augment the original matrix \( H \) with an identity matrix of the same size. This looks like \( [H | I] \), where \( H \) is on the left and the identity matrix \( I \) is on the right.
This joins the two matrices into an augmented matrix, enabling us to apply operations easily and see real-time transformations on both sides. All changes made to the matrix on the left will reflect potential transformations needed on the identity, which eventually guides us toward revealing the inverse, if it exists.

Gauss-Jordan Elimination

Gauss-Jordan elimination is a method used to solve systems of linear equations. It's also central to finding the inverse of a matrix.
The objective is to transform the original matrix into the reduced row-echelon form, which should ideally convert the left side of the augmented matrix into an identity matrix. This is achieved by using simple operations:

• Switching two rows
• Multiplying a row by a non-zero constant
• Adding or subtracting multiples of rows

These operations tweak the matrix step by step until it's simplified. If you can turn the left side into the identity matrix, the transformations will reveal what \( H^{-1} \) looks like on the other side of the augmented matrix.
However, if you reach a point where certain steps can't make the identity matrix form, it's evident that the original matrix isn't invertible.

Identity Matrix

The identity matrix is an important concept in linear algebra. It is a square matrix that serves as the multiplicative identity for matrix multiplication. This means that for any matrix \( A \), when multiplied with the identity matrix \( I \), you get \( A \) back (\( A \times I = A \) and \( I \times A = A \)).
The identity matrix contains 1s on the main diagonal and 0s elsewhere. For example, for a 3x3 identity matrix, it looks like this:\[I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
While working on matrix inversion, the identity matrix is the target; you wish to transform the left side of the augmented matrix into this identity form. Successfully doing so indicates that the inverse exists and is found on the other side of the augmented matrix.
In the exercise you are working on, transforming the original matrix completely into an identity on the left side failed. This indicated that an inverse is not possible for the given matrix.