Chapter 7: Problem 14
Either compute the inverse of the given matrix, or else show that it is singular. \(\left(\begin{array}{rrr}{1} & {2} & {1} \\ {-2} & {1} & {8} \\ {1} & {-2} & {-7}\end{array}\right)\)
Short Answer
Expert verified
Question: Determine if the given matrix is singular or not. If not, find its inverse:
Matrix:
\begin{equation}
\left(\begin{array}{ccc}{1} & {2} & {1} \\\ {-2} & {1} & {8} \\\ {1} & {-2} & {-7}\end{array}\right)
\end{equation}
Answer: The given matrix is not singular since its determinant is 19. Its inverse is:
Inverse:
\begin{equation}
\left(\begin{array}{ccc}{-\frac{23}{19}} & {\frac{30}{19}} & {\frac{2}{19}} \\\ {-\frac{22}{19}} & {\frac{8}{19}} & {\frac{22}{19}} \\\ {0} & {-\frac{4}{19}} & {-\frac{3}{19}}\end{array}\right)
\end{equation}
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant of a Matrix
The determinant of a matrix is a scalar value that is a fundamental property used in matrix theory. It tells us whether a square matrix is invertible or not, amongst other things. A singular matrix, one that does not have an inverse, will have a determinant of zero. To calculate the determinant of a 3x3 matrix, we use a method that involves cross-multiplication of the rows and columns, excluding the row and column of the current element. This method is essentially a sum of products of diagonals (positive and negative).
For example, for a matrix A, the determinant is calculated using the formula:
\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]
Where \(a_{ij}\) denotes the element in the i-th row and j-th column of matrix A. If this calculation yields any non-zero value, the matrix has an inverse; if it's zero, the matrix is singular and inverses cannot be found.
For example, for a matrix A, the determinant is calculated using the formula:
\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]
Where \(a_{ij}\) denotes the element in the i-th row and j-th column of matrix A. If this calculation yields any non-zero value, the matrix has an inverse; if it's zero, the matrix is singular and inverses cannot be found.
Cofactor Matrix
A cofactor matrix is an intermediate step in computing the inverse of a matrix. To find the cofactors, you must first calculate the minors of the matrix. Each minor is the determinant of a smaller matrix that you get by removing the i-th row and j-th column from the original matrix. The cofactor is obtained by applying a checkerboard pattern of plus and minus signs to these minors, known as the sign factor, given by \((-1)^{i+j}\) where \(i\) and \(j\) are row and column indices respectively.
The matrix of cofactors is simply the original matrix with each element replaced by its cofactor. This process reflects the position's influence on the determinant and entails coupling each minor with the appropriate sign.
The matrix of cofactors is simply the original matrix with each element replaced by its cofactor. This process reflects the position's influence on the determinant and entails coupling each minor with the appropriate sign.
Adjoint Matrix
Once we have obtained the cofactor matrix, we can find the adjoint of the original matrix. The adjoint, or adjugate, of a matrix is the transpose of its cofactor matrix. Transposing a matrix involves flipping it over its diagonal, which means writing the rows of the original matrix as columns.
The adjoint plays a key role in finding the inverse because the inverse is calculated by dividing the adjoint by the original matrix's determinant. The essence of using the adjoint in inversion is that it represents the unique combination of row operations that transform the original matrix into the identity matrix when multiplied by \(1/\text{det}(A)\).
The adjoint plays a key role in finding the inverse because the inverse is calculated by dividing the adjoint by the original matrix's determinant. The essence of using the adjoint in inversion is that it represents the unique combination of row operations that transform the original matrix into the identity matrix when multiplied by \(1/\text{det}(A)\).
Singular Matrix
A singular matrix is a square matrix that does not have an inverse. In terms of computation, it is identified by its determinant. If the determinant of a matrix is zero, then the matrix is singular. This indicates that there is no unique solution to a system of linear equations represented by this matrix.
Singular matrices cannot be used in division and therefore pose problems in various matrix operations, including solving systems of equations or finding eigenvalues. Recognizing a singular matrix before attempting inversion is crucial to avoid computational errors and to understand the limitations of a given set of linear equations.
Singular matrices cannot be used in division and therefore pose problems in various matrix operations, including solving systems of equations or finding eigenvalues. Recognizing a singular matrix before attempting inversion is crucial to avoid computational errors and to understand the limitations of a given set of linear equations.
