Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 5 & 4 \\ 3 & 6 & 5 \end{array}\right] $$

Short answer

The inverse of the given matrix is \[\begin{array}{lll} 1 & -2 & 1 \ -2 & 1 & 1 \ 1 & 1 & 1 \end{array}\right]\n\]

Step-by-step solution

Compute the determinant

First, we need to compute the determinant of the given matrix. We do this by expanding along the first row:\n \[1(5 \cdot 5 - 4 \cdot 6) - 1(3 \cdot 5 - 4 \cdot 3) + 1(3 \cdot 6 - 5 \cdot 3)\] which simplifies to 1.

Compute the matrix of cofactors

Next, compute the matrix of cofactors. This is done by taking the determinant of every 2x2 submatrix, and then multiplying every alternate determinant by -1.\n The matrix of cofactors is:\n\[\begin{array}{lll} 1 & -2 & 1 \ -2 & 1 & 1 \ 1 & 1 & 1 \end{array}\right]\n\]

Compute the adjugate

Then, compute the adjugate (also called adjoint) which is the transpose of the cofactor matrix.\n The adjugate is:\n\[\begin{array}{lll} 1 & -2 & 1 \ -2 & 1 & 1 \ 1 & 1 & 1 \end{array}\right]\n\]

Compute the inverse

Finally, the inverse of the original matrix is obtained by dividing every element in the adjugate by the determinant of the original matrix. As the determinant was calculated to be 1, the inverse matrix is equal to the adjugate:\n\[\begin{array}{lll} 1 & -2 & 1 \ -2 & 1 & 1 \ 1 & 1 & 1 \end{array}\right]\n\]

Key concepts

Understanding the Determinant

The determinant of a matrix is a scalar value that offers insights into the matrix properties, like whether it has an inverse. To find a determinant for a 3x3 matrix, we expand along one row or column. Here, expanding along the first row, we apply the formula:

• Choose elements of the row: 1, 1, 1.
• Calculate the determinants of the corresponding 2x2 submatrices.
• Multiply each by their respective signs, based on their position, using: "+,-,+".

The result is a scalar value of 1. Since it is non-zero, an inverse exists.

Exploring the Adjugate

The adjugate, or adjoint, of a matrix, is crucial for finding the inverse. It is obtained by transposing the cofactor matrix. The process is:

• Start with the cofactor matrix.
• Switch rows and columns to transpose.

For our matrix, the adjugate remains the same after transposition. This is because the cofactor matrix is symmetrical, meaning the adjugate is:\[\begin{array}{lll}1 & -2 & 1 \-2 & 1 & 1 \1 & 1 & 1\end{array}\]

Building the Cofactor Matrix

The cofactor matrix is composed by finding minors and applying signs. Here's how it's created:

• Identify each element's submatrix by removing the row and column it belongs to.
• Calculate the determinant of each submatrix.
• Apply the checkerboard pattern of signs: "+,-,+" for the first row, "-,+, -" for the second, and "+,-,+" for the third.

Thus, the elements form our cofactor matrix:\[\begin{array}{lll}1 & -2 & 1 \-2 & 1 & 1 \1 & 1 & 1\end{array}\]These steps ensure each element correctly contributes to finding the adjugate.

Significance of the 2x2 Submatrix

2x2 submatrices are smaller matrices formed by omitting a row and a column from the original matrix. They are critical for computing cofactors.

• For a 3x3 matrix, each element is associated with a 2x2 submatrix.
• Find the determinant of each by multiplying diagonally: top-left to bottom-right minus top-right to bottom-left.
• The 2x2 submatrices simplify the process of finding larger matrix determinants.

By understanding 2x2 submatrices, you learn the foundation of more complex determinant calculations.