Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 1 & 0 & 0 \\ 3 & 4 & 0 \\ 2 & 5 & 5 \end{array}\right] $$

Short answer

The inverse of the matrix is \(\left[\begin{array}{lll} 1/20 & 0 & 0 \ 3/20 & 1/5 & 0 \ 2/20 & 1/4 & 1/4 \end{array}\right]\)

Step-by-step solution

Check if inverse exists

Calculate the determinant of the matrix. The determinant of the given 3x3 matrix is \(1(4*5-0) - 0 - 0 = 20\). Because the determinant is not zero, an inverse of the matrix exists.

Find the matrix of minors

Determine the matrix of minors. This involves replacing each element in the matrix with the determinant of the sub-matrix that is left when the row and column containing that element is removed. Given that it's a diagonal matrix, the matrix of minors is the matrix itself.

Apply Cofactor

Find the cofactor matrix. This involves applying a checkerboard of minuses to the 'Matrix of Minors'. As the given matrix is a diagonal one, and so functioned as its matrix of minors, the cofactor matrix is also the same as the original matrix \(\left[\begin{array}{lll} 1 & 0 & 0 \ 3 & 4 & 0 \ 2 & 5 & 5 \end{array}\right]\)

Find the adjugate

Find the adjugate, which is the transpose of the cofactor matrix. For the given matrix, as for any diagonal matrix, the adjugate is the matrix itself.

Compute the Inverse

Lastly, the inverse of the matrix is computed by dividing every element of the adjugate matrix by the determinant. Thus the inverse matrix of the given matrix is \(\left[\begin{array}{lll} 1/20 & 0 & 0 \ 3/20 & 1/5 & 0 \ 2/20 & 1/4 & 1/4 \end{array}\right]\)

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It is crucial because it helps us decide if a matrix has an inverse. For a 3x3 matrix, you calculate the determinant by a specific formula that involves the elements and their cross products. Given a 3x3 matrix:\[\left[\begin{array}{ccc}a & b & c \d & e & f \g & h & i \\end{array}\right]\]the determinant is computed as:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]In the exercise, the determinant was calculated as 20. Since this number wasn't zero, it indicates that the matrix is invertible. A zero determinant means the matrix can't be inverted, which makes understanding and calculating it foundational in finding inverses.

Matrix of Minors

The matrix of minors is a matrix where each element is replaced by the determinant of a smaller matrix. This smaller matrix is obtained by removing the row and column of the current element from the original matrix. This step helps in finding the cofactor matrix.For example, if we start with:\[\left[\begin{array}{ccc}1 & 0 & 0 \3 & 4 & 0 \2 & 5 & 5 \\end{array}\right]\]the minors are determined by calculating the determinant of each 2x2 matrix that remains after removing the respective row and column. However, for some matrices, like the one in the exercise, the matrix of minors can be the matrix itself if it has certain properties, such as being a diagonal matrix. This simplifies the process significantly.

Cofactor Matrix

The cofactor matrix is generated by applying a pattern of positive and negative signs to the matrix of minors. This pattern creates an arrangement that looks like a checkerboard of plus and minus signs.

• The top left element stays positive,
• the next element in the row changes its sign,
• and so on.

This alternation is crucial because it allows for the construction of the adjugate matrix later.In matrix terms, each element in the cofactor matrix is calculated as:\[(-1)^{i+j} M_{ij}\]Where \(i\) and \(j\) represent the row and column of the element, and \(M_{ij}\) is the corresponding minor. In the exercise's case with a diagonal matrix, the cofactor matrix remains unchanged, simplifying our calculations.

Adjugate Matrix

The adjugate matrix is the transpose of the cofactor matrix. Transposing a matrix involves swapping its rows and columns. This step is essential in calculating the inverse of the matrix.Consider the same 3x3 matrix:\[\left[\begin{array}{ccc}1 & 0 & 0 \3 & 4 & 0 \2 & 5 & 5 \\end{array}\right]\]In the exercise, since it's a diagonal matrix, its transpose results in the same matrix. Thus, the adjugate matrix is also the original matrix. This property greatly eases calculations because it avoids any additional modifications before finding the inverse. Once the adjugate is established, each element is divided by the determinant to determine the inverse of the original matrix.