Question

Use a calculator or computer to compute \(A A^{-1},\) where $$ A=\left[\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 1 & 4 & 9 & 16 \\ 1 & 8 & 27 & 64 \\ 1 & 16 & 81 & 256 \end{array}\right] $$ Was the identity matrix returned exactly? Comment on your results.

Short answer

The resulting matrix should ideally be the identity matrix. Any deviation could be due to numerical precision limits.

Step-by-step solution

Definition of Matrix and Inverse

Matrix multiplication of a matrix with its inverse, denoted as \(AA^{-1}\), should yield the identity matrix, \(I\), if \(A\) is invertible. The identity matrix for a 4x4 matrix is \(\begin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{bmatrix}\).

Compute the Inverse of Matrix A

Use a calculator or computer software to calculate the inverse of matrix \(A\). Ensure proper input of matrix values to acquire \(A^{-1}\), which is typically done through matrix operations functions such as `inv()` or equivalent on the chosen calculator/software.

Perform Matrix Multiplication

Multiply the matrix \(A\) by its calculated inverse \(A^{-1}\) using a calculator or computer software. This multiplication should be handled via built-in functions to ensure accuracy, especially when dealing with larger matrices.

Check for Identity Matrix

Examine the result of the multiplication from Step 3. In the ideal scenario, \(AA^{-1}\) should exactly equal the 4x4 identity matrix. Check each element to see if it closely matches, considering any potential numerical approximations or rounding errors that might occur when using calculators or computers.

Key concepts

Inverse Matrix

An inverse matrix is a special entity in matrix algebra. When you multiply a matrix with its inverse, you get an identity matrix. This is similar to how multiplying a number by its reciprocal equals one, like 3 times 1/3 equals 1.
Just like for regular numbers, not every matrix has an inverse. Only square matrices (matrices with the same number of rows and columns) might have an inverse, and even then, it must be a "full rank" matrix. This means all its rows (or columns) should be independent.
The process of finding the inverse of a matrix typically involves various methods. These include the Gauss-Jordan elimination, or using special computer functions such as `inv()` in programming languages. These methods calculate what changes are needed to transform the matrix into an identity matrix.

Matrix Multiplication

Matrix multiplication involves a series of dot products between the rows of the first matrix and the columns of the second.
This operation isn't as straightforward as multiplying regular numbers.
Here are some key points about matrix multiplication:

• The number of columns in the first matrix must equal the number of rows in the second.
• The product matrix dimensions are determined by the rows of the first and the columns of the second matrix.
• Order matters. In general, matrices are not commutative; that is, if you multiply matrix A by matrix B, it might not be the same as multiplying matrix B by matrix A.

For the example in the exercise, since we are multiplying a matrix by its inverse, the result should be an identity matrix where possible.

Identity Matrix

An identity matrix acts like the number 1 in matrix world. When you multiply any matrix by the identity matrix, it stays unchanged.
Essentially, it consists of 1s down its main diagonal from the top left to the bottom right, and 0s everywhere else. For instance, a 4x4 identity matrix looks like this:
\[\begin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{bmatrix}\]
It's important because when you multiply a matrix by its inverse, the result should ideally be an identity matrix, confirming that the original and its inverse are correct inverses of each other.

Numerical Approximation

Numerical approximation is important, especially when working with real-life data or calculating on a computer or calculator.
Due to the nature of floating-point arithmetic, results can sometimes contain small errors. These are a result of how numbers are stored in binary on these devices.
When computing matrix inversions or multiplications in such environments, you may find the result isn't an exact identity matrix but very close. The presence of tiny discrepancies like 0.0001 instead of 0 might occur.

• These tiny errors are generally acceptable and expected in most practical applications.
• Understanding these approximations is key to comprehending the potential small "errors" in computational results.
• It's also crucial to interpret such results accurately rather than expecting perfect precision, which is mostly infeasible.

So in problems like the one described in the exercise, the finished product might be slightly off from the perfect identity matrix due to these approximations.