Question

Either compute the inverse of the given matrix, or else show that it is singular. \(\left(\begin{array}{lll}{2} & {1} & {0} \\ {0} & {2} & {1} \\ {0} & {0} & {2}\end{array}\right)\)

Short answer

Question: Compute the inverse of the given 3x3 matrix or show that it is singular: \(A=\left(\begin{array}{lll}{2} & {1} & {0} \\\ {0} & {2} & {1} \\\ {0} & {0} & {2}\end{array}\right)\) Answer: The inverse of the given matrix is \(A^{-1}=\left(\begin{array}{lll}{\frac{1}{2}} & {0} & {0} \\\ {-\frac{1}{8}} & {\frac{1}{2}} & {0} \\\ {0} & {-\frac{1}{8}} & {\frac{1}{2}}\end{array}\right)\)

Step-by-step solution

Compute the determinant of the given matrix

We are given the matrix: \(A=\left(\begin{array}{lll}{2} & {1} & {0} \\\ {0} & {2} & {1} \\\ {0} & {0} & {2}\end{array}\right)\) Since it is a triangular matrix, we can compute the determinant as the product of the diagonal elements: Det(A) = 2 * 2 * 2 = 8 Since the determinant is not zero, the matrix is not singular, and we can compute its inverse.

Compute the matrix of cofactors

First, we will compute the matrix of cofactors of matrix A: Cof(A) = \(\left(\begin{array}{lll}{C_{11}} & {C_{12}} & {C_{13}} \\\ {C_{21}} & {C_{22}} & {C_{23}} \\\ {C_{31}} & {C_{32}} & {C_{33}}\end{array}\right)\) For each element, we can calculate its cofactor as: \(C_{ij} = (-1)^{i+j} M_{ij}\) Where \(M_{ij}\) is the minor of element \(a_{ij}\). Since matrix A is a triangular matrix, the cofactors of the upper and lower triangle are easy to compute: Cof(A) = \(\left(\begin{array}{lll}{4} & {-1} & {0} \\\ {0} & {4} & {-1} \\\ {0} & {0} & {4}\end{array}\right)\)

Compute the adjugate of the matrix of cofactors

Now we need to compute the adjugate of the matrix of cofactors. The adjugate is the transpose of the matrix of cofactors: Adj(A) = \(\left(\begin{array}{lll}{4} & {0} & {0} \\\ {-1} & {4} & {0} \\\ {0} & {-1} & {4}\end{array}\right)\)

Compute the inverse of matrix A

Finally, we can compute the inverse of matrix A by multiplying the adjugate of the matrix of cofactors by the reciprocal of the determinant: \(A^{-1}=\frac{1}{\text{det}(A)} \cdot \text{Adj}(A) = \frac{1}{8} \left(\begin{array}{lll}{4} & {0} & {0} \\\ {-1} & {4} & {0} \\\ {0} & {-1} & {4}\end{array}\right)\) So the inverse of matrix A is: \(A^{-1}=\left(\begin{array}{lll}{\frac{1}{2}} & {0} & {0} \\\ {-\frac{1}{8}} & {\frac{1}{2}} & {0} \\\ {0} & {-\frac{1}{8}} & {\frac{1}{2}}\end{array}\right)\)

Key concepts

Determinant

The determinant of a matrix is a special number that can be calculated from its elements. It tells us important information about the matrix, such as whether the matrix is invertible or singular. If the determinant is zero, the matrix is singular and does not have an inverse. If it's not zero, the matrix is invertible.

For triangular matrices, calculating the determinant becomes a simpler task. A triangular matrix is a type where all elements above or below the main diagonal are zero. In our case, the matrix given is upper triangular. Calculating the determinant of a triangular matrix requires multiplying all the diagonal elements together.

So for the matrix \(A\) given, we have:

• Diagonal elements: 2, 2, 2

Multiply these together to find the determinant:

• \( ext{det}(A) = 2 \times 2 \times 2 = 8 \)

This shows us that the determinant is not zero, indicating that the matrix is invertible.

Triangular Matrix

A triangular matrix is a kind of matrix with zeros in all spaces either above or below the main diagonal. We can classify them as either upper or lower triangular matrices. Upper triangular matrices have non-zero entries only on or above the main diagonal, while lower triangular matrices have non-zero entries below the diagonal.

In the problem provided, we have an upper triangular matrix:

• \(\left(\begin{array}{ccc}2 & 1 & 0 \0 & 2 & 1 \0 & 0 & 2\end{array}\right)\)

This structure simplifies many matrix operations, such as determinant calculation, and helps in solving systems of equations along with finding inverses, making them quite handy in computational mathematics.

Matrix of Cofactors

The matrix of cofactors is an intermediate step in finding the inverse of a matrix. To construct this matrix, we need to compute the cofactor for each element in the matrix.

The cofactor, \( C_{ij} \), of an element \( a_{ij} \) is determined by:

• First finding the minor, \( M_{ij} \), by removing the ith row and jth column from the original matrix.
• Applying the formula \( C_{ij} = (-1)^{i+j} M_{ij} \).

For our matrix:\( \begin{pmatrix}2 & 1 & 0 \0 & 2 & 1 \0 & 0 & 2\end{pmatrix} \), the cofactors result in the matrix:

• \(\left(\begin{array}{ccc}4 & -1 & 0 \0 & 4 & -1 \0 & 0 & 4 \end{array}\right)\)

This matrix of cofactors will be used to find the adjugate, which is a step closer to the matrix inverse.

Adjugate

The adjugate of a matrix is essentially the transpose of its cofactor matrix. This is a crucial step in determining the inverse of a given matrix.

To find the adjugate, replace each element of the matrix with its corresponding cofactor, and then transpose the resulting matrix.

For the given matrix:

• Cofactor Matrix: \(\begin{pmatrix}4 & -1 & 0 \0 & 4 & -1 \0 & 0 & 4 \end{pmatrix} \)
• Transpose of Cofactor Matrix (Adjugate): \(\begin{pmatrix}4 & 0 & 0 \-1 & 4 & 0 \0 & -1 & 4 \end{pmatrix} \)

The adjugate matrix is then used in the final step to compute the inverse by multiplying it by the reciprocal of the determinant of the original matrix.