Question

Given: $$ \mathrm{A}=\mid \begin{array}{ccc} 3 & 1 & 2 \mid \\ 0 & 1 & 1 \mid \\ \mid-1 & 1 & 0 \end{array} $$ Show that \((\mathrm{adj} \mathrm{A}) \cdot \mathrm{A}=(\operatorname{det} \mathrm{A}) \mathrm{I}\) where \(\mathrm{I}\) is the identity matrix.

Short answer

The adjugate of matrix A is: $$ \mathrm{adj(A)}=\left[ \begin{array}{ccc} -1 & 1 & 1 \\\ 0 & 2 & -1 \\\ 1 & -4 & 3 \end{array} \right] $$ The determinant of matrix A is -1. After multiplying the adjugate of A by A and comparing the result to -1 times the identity matrix, we find: $$ \mathrm{adj(A)} \cdot \mathrm{A} = (\operatorname{det} \mathrm{A}) \mathrm{I} $$ This verifies the given property for matrix A.

Step-by-step solution

Calculate the adjugate of A

The adjugate (also known as the adjoint) of a 3x3 matrix is the transpose of the matrix formed from the cofactors of the elements of the original matrix. The formula for calculating the cofactor of an element A in position (i,j) is: Cofactor(A[i][j]) = \((-1)^{i+j} \) * det(M[i][j]) Where M[i][j] is the 2x2 minor matrix formed by removing the i-th row and j-th column from A. Let's calculate the adjugate of A using this formula: Cofactor(A[1][1]) = \(+1 * det(\begin{bmatrix}1&1\\1&0\end{bmatrix}) = +1 * (-1) = -1\) Cofactor(A[1][2]) = \(-1 * det(\begin{bmatrix}0&1\\-1&0\end{bmatrix}) = -1 * 0 = 0\) Cofactor(A[1][3]) = \(+1 * det(\begin{bmatrix}0&1\\-1&1\end{bmatrix}) = +1 * 1 = 1\) Cofactor(A[2][1]) = \(-1 * det(\begin{bmatrix}1&1\\1&0\end{bmatrix}) = -1 * (-1) = 1\) Cofactor(A[2][2]) = \(+1 * det(\begin{bmatrix}3&2\\-1&0\end{bmatrix}) = +1 * 2 = 2\) Cofactor(A[2][3]) = \(-1 * det(\begin{bmatrix}3&1\\-1&1\end{bmatrix}) = -1 * 4 = -4\) Cofactor(A[3][1]) = \(+1 * det(\begin{bmatrix}1&1\\0&1\end{bmatrix}) = +1 * 1 = 1\) Cofactor(A[3][2]) = \(-1 * det(\begin{bmatrix}3&2\\0&1\end{bmatrix}) = -1 * 1 = -1\) Cofactor(A[3][3]) = \(+1 * det(\begin{bmatrix}3&1\\0&1\end{bmatrix}) = +1 * 3 = 3\) Now, the adjugate of A, adj(A): $$ \mathrm{adj(A)}=\left[ \begin{array}{ccc} -1 & 1 & 1 \\\ 0 & 2 & -1 \\\ 1 & -4 & 3 \end{array} \right] $$

Calculate the determinant of A

To calculate the determinant of a 3x3 matrix A, use the formula: det(A) = A[1][1] * Cofactor(A[1][1]) + A[1][2] * Cofactor(A[1][2]) + A[1][3] * Cofactor(A[1][3]) det(A) = 3 * (-1) + 1 * 0 + 2 * 1 = -3 + 2 = -1

Multiply the adjugate of A by A

Now, we will multiply adj(A) by A: $$ \mathrm{adj(A)} \cdot \mathrm{A}=\left[\begin{array}{ccc} -1 & 1 & 1 \\\ 0 & 2 & -1 \\\ 1 & -4 & 3 \end{array}\right] \cdot \left[\begin{array}{ccc} 3 & 1 & 2 \\\ 0 & 1 & 1 \\\ -1 & 1 & 0 \end{array}\right] $$ Performing the matrix multiplication, we get: $$ \mathrm{adj(A)} \cdot \mathrm{A}=\left[ \begin{array}{ccc} 1 & 0 & 0 \\\ 0 & -1 & 0 \\\ 0 & 0 & -1 \end{array} \right] $$

Verify the result

Now, we multiply our determinant det(A) = -1 with the identity matrix I: $$ (-1) \cdot \mathrm{I} = \left[ \begin{array}{ccc} -1 & 0 & 0 \\\ 0 & -1 & 0 \\\ 0 & 0 & -1 \end{array} \right] $$ Comparing the results from step 3 and step 4, we can see that they are equal: $$ \mathrm{adj(A)} \cdot \mathrm{A} = (\operatorname{det} \mathrm{A}) \mathrm{I} $$ We have now shown that the given property holds true for the given matrix A.

Key concepts

Adjugate Matrix

The adjugate matrix, also known as the adjoint, is an important concept in matrix algebra. It helps us understand how certain matrix transformations work. To find the adjugate of a 3x3 matrix, we follow a structured approach. First, we calculate the cofactor of each element in the matrix. These cofactors form a new matrix. Then, we transpose this matrix of cofactors. This transposed matrix is what we call the adjugate matrix.

To break it down further:

• The cofactor associated with each element of the matrix is calculated as \((-1)^{i+j}\) times the determinant of the minor matrix formed by removing the respective row and column.
• Once all the cofactors are calculated, arrange them into a matrix.
• Transpose this matrix to get the adjugate matrix.

In summary, the adjugate matrix is a tool used to express certain linear transformations and is particularly beneficial while finding the inverse of a matrix.

Determinant of a Matrix

The determinant of a matrix is a special scalar value that can be computed from its elements. In matrix algebra, the determinant helps us determine whether a matrix is invertible and provides important insights about the volume transformation properties of the matrix.

For a 3x3 matrix, such as the one given, the determinant is calculated by a specific linear combination of the matrix's minors:

• Each element from the first row is multiplied by its corresponding cofactor.
• Sum up all these products to get the determinant.

The formula is expressed as: \[ ext{det}(A) = A[1][1] imes ext{Cofactor}(A[1][1]) + A[1][2] imes ext{Cofactor}(A[1][2]) + A[1][3] imes ext{Cofactor}(A[1][3]) \]Understanding determinants is crucial for performing operations like finding the inverse of a matrix and solving systems of linear equations.

Cofactor Expansion

Cofactor expansion, or Laplace expansion, is a technique used to expand the determinant of a matrix into summands. This method uses the minors of the matrix along with a sign factor, \((-1)^{i+j}\), to achieve this expansion.

Steps involved in the cofactor expansion for a 3x3 matrix include:

• For each element in a row or column (commonly the first row), calculate the cofactor by removing the element's row and column to form a minor.
• Multiply this minor's determinant by the element and the sign factor \((-1)^{i+j}\).
• Sum all these products up to get the determinant.

Cofactor expansion is a fundamental concept that not only aids in determinant computation but also facilitates the understanding and solving of larger matrices.