Question

Either compute the inverse of the given matrix, or else show that it is singular. \(\left(\begin{array}{lll}{1} & {2} & {3} \\ {2} & {4} & {5} \\ {3} & {5} & {6}\end{array}\right)\)

Short answer

Answer: The inverse of the matrix is $ \textbf{A}^{-1} = \begin{pmatrix} \frac{1}{7} & -\frac{1}{7} & -\frac{2}{7} \\ -\frac{1}{7} & \frac{1}{7} & \frac{3}{7} \\ 0 & \frac{2}{7} & -\frac{3}{7} \end{pmatrix} $.

Step-by-step solution

Compute the determinant of the matrix

To check if the given matrix is invertible, we need to compute its determinant. We're given this matrix: $ \textbf{A}=\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix} $ Computing its determinant, we get: $ \det(\textbf{A}) = 1(24 - 25) - 2(12 - 15) + 3(10 - 10) = (-1) + (-6) + 0 = -7 $ The determinant is non-zero, which means the matrix has an inverse.

Compute the matrix of minors

We compute the matrix of minors by finding the determinant of the 2x2 matrix that results from removing the row and column of each element. The matrix of minors for \(\textbf{A}\) is: $ \textbf{M} = \begin{pmatrix} -1 & (-1) & 0 \\ 1 & 1 & 2 \\ 2 & 3 & 3 \end{pmatrix} $

Compute the matrix of cofactors

Next, we compute the matrix of cofactors by applying a checkerboard pattern of signs to the matrix of minors. The matrix of cofactors for \(\textbf{A}\) is: $ \textbf{C} = \begin{pmatrix} -1 & 1 & 0 \\ 1 & -1 & -2 \\ 2 & -3 & 3 \end{pmatrix} $

Compute the adjugate of the matrix

The adjugate of a matrix is simply its transpose. We'll transpose the matrix of cofactors to get the adjugate: $ \textbf{adj}(\textbf{A}) = \begin{pmatrix} -1 & 1 & 2 \\ 1 & -1 & -3 \\ 0 & -2 & 3 \end{pmatrix} $

Compute the inverse of the matrix

Finally, we divide the adjugate by the determinant to find the inverse of the matrix: $ \textbf{A}^{-1} = \frac{1}{\det(\textbf{A})} \times \textbf{adj}(\textbf{A}) = \frac{1}{-7} \begin{pmatrix} -1 & 1 & 2 \\ 1 & -1 & -3 \\ 0 & -2 & 3 \end{pmatrix} $ Our final answer is the matrix inverse: $ \textbf{A}^{-1} = \begin{pmatrix} \frac{1}{7} & -\frac{1}{7} & -\frac{2}{7} \\ -\frac{1}{7} & \frac{1}{7} & \frac{3}{7} \\ 0 & \frac{2}{7} & -\frac{3}{7} \end{pmatrix} $

Key concepts

Determinant of a Matrix

When exploring the concept of matrix inversion, the determinant of a matrix is a crucial starting point. Understanding the determinant is the key to determining whether a square matrix is invertible or singular. Physically, the determinant can be thought of as a scaling factor that a matrix applies to the space it operates on. Mathematically, for a 2x2 matrix, it is computed as the difference between the products of its diagonals. For larger matrices, the calculation involves a more intricate process which might include the use of minors and cofactors, but ultimately results in a single scalar value.

The determinant not only decides the invertibility of a matrix but also plays a role in various mathematical and applied contexts, such as solving systems of linear equations and evaluating eigenvalues. If the determinant is zero, the matrix cannot be inverted, indicating that, in terms of transformations, it squishes the space into a smaller dimension, losing information that cannot be retrieved. Conversely, a non-zero determinant, as in the exercise provided, ensures the existence of an inverse matrix.

Matrix of Minors

The next step in finding a matrix inverse involves the matrix of minors. A minor of a matrix is determined for each element of the matrix and is the determinant of a smaller matrix that's left when the row and column containing that element are deleted. To form the matrix of minors, we calculate these minors across all positions of the original matrix.

In doing so, you piece together a new matrix where each element corresponds to the determinant of its minor. For larger matrices, this process becomes more computationally intense, highlighting elements' interconnectedness within the matrix. The concept of minors is fundamental to not only inverting matrices but also deeper algebraic properties, such as Cramer’s Rule and calculating determinants using Laplace's Expansion. When looking at the provided example, the calculation of each minor reflects how the removal of certain rows and columns can influence the overall matrix.

Matrix of Cofactors

Once the matrix of minors is established, we proceed to the matrix of cofactors. A cofactor includes a sign change contingent on the element's position in the matrix. Using a checkerboard pattern of positives and negatives—starting with a positive at the top-left corner—each minor is multiplied by either +1 or -1 accordingly. This sign modification is mathematically represented as \( (-1)^{i+j} \) times the minor of the element in the i-th row and j-th column.

The resulting matrix of cofactors carries the same dimensions as the original matrix and is an essential intermediary step in finding the adjugate and the inverse. This adjustment of signs is a vital element in ensuring that the reconstructed vector space, after matrix inversion, properly aligns with the orientation and transformation that the original matrix would induce.

Adjugate of a Matrix

The final piece of the puzzle before we arrive at the inverse is the adjugate of a matrix. The adjugate, or classical adjoint, is the transpose of the matrix of cofactors. By transposing, we swap the matrix's rows and columns. This operation reflects the importance of the relative positions of elements within the matrix, providing a structure that complements the original matrix when calculating the inverse.

For our given matrix, the transpose of the matrix of cofactors gives us the adjugate. This step maintains the geometric interpretations held by cofactors, essentially re-orienting them to allow division by the determinant to return a matrix that reflects the original matrix's 'opposite' operation—its inverse. When the adjugate is combined with the reciprocal of the determinant in the final inversion formula, the resulting inverse matrix holds all the secrets to undoing the transformations applied by the original matrix.