Question

Inverse of \(\left[\begin{array}{rrr}1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1\end{array}\right]\) is \(\left[\begin{array}{lll}3 & 2 & 6 \\ 1 & 1 & k \\ 2 & 2 & 5\end{array}\right]\) then \(k\) is .......

Short answer

k = 0

Step-by-step solution

Understand the Concept of Matrix Inversion

The inverse of a matrix, when multiplied by the original matrix, gives the identity matrix. For a 3x3 matrix \(A\), let's assume the product \(A \cdot A^{-1} = I_3\), where \(I_3\) is the identity matrix of size 3x3.

Multiply the Given Matrices

Calculate the product of the given matrix and its purported inverse. Multiply each row of the first matrix with each column of the second matrix, and sum the products for each element.

Set Up the Product as Identity Matrix

Ensure that the resulting matrix from the multiplication becomes the identity matrix. Use the equation form obtained from multiplication to equate it to the identity matrix and solve for \(k\).

Solve for k

After performing the matrix multiplication, equate the derived expressions in the resulting matrix to the identity matrix. Particularly, focus on the (2,2) position of the resulting product matrix and solve for \(k\).

Key concepts

Inverse of a Matrix

The inverse of a matrix is like a magical undo button for matrices. Imagine you have a number, and its inverse is like the reciprocal. For numbers, this means when you multiply a number by its inverse, you get 1. Similarly, with matrices, when you multiply a matrix by its inverse, the identity matrix pops up! But not all matrices have inverses. A matrix must be square (same number of rows and columns) and must be "non-singular," which means it must have a non-zero determinant.
To find the inverse of a 3x3 matrix, mathematicians use various methods, including the adjugate method or Gaussian elimination. In this case, we need to take an additional step and multiply it back to the original matrix to ensure it gives the identity matrix.
This verification step is crucial because it's possible to make errors when computing the inverse manually. Checking via multiplication ensures you have the correct inverse.

Identity Matrix

Picture the identity matrix as the number 1 in matrix form. Just like how multiplying any number by 1 leaves it unchanged, multiplying any matrix by the identity matrix leaves it the same. The identity matrix is a special kind of square matrix made up entirely of 1's on the diagonal and 0's elsewhere.
For a 3x3 matrix, the identity matrix looks like this:

• Diagonal values: 1, 1, 1
• All other values: 0

When you perform matrix multiplication, if the result doesn't look like an identity matrix, it means there's a mistake somewhere. In the exercise, when we multiply the given 3x3 matrix with its inverse, our goal is to achieve the identity matrix. If not achieved, it suggests that the assumed inverse or computation needs re-examination.

Matrix Multiplication

Matrix multiplication is a process combining rows and columns to get a new matrix. Unlike regular multiplication, order matters! If you have two matrices, say A and B, multiplying A by B isn't the same as multiplying B by A.
This operation involves multiplying each element of the row of the first matrix with each element of the column of the second matrix and then adding those products together to form a new element in the resultant matrix. This is why understanding matrix multiplication's nature is important, especially when finding the inverse.
In this exercise, the original matrix is multiplied by its proposed inverse. Each position in the resulting matrix tells us if our proposed inverse is correct. For example, multiplying the second row of the original matrix with the second column of the inverse matrix helps us find the values to solve for, like the variable \( k \). If correctly set up, it matches the corresponding position in the identity matrix.