Question

Show that adj \((u A)=u^{n-1}\) adj \(A\) for all \(n \times n\) matrices \(A\)

Short answer

adj \\(u A) = u^{n-1}\\ adj \\(A) follows from scaling properties of determinants and adjugates.

Step-by-step solution

Understand the Concept of an Adjugate Matrix

The adjugate (or adjoint) of a square matrix \(A\), denoted by \(\text{adj}(A)\), is the transpose of its cofactor matrix. It satisfies the equation \(A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \text{det}(A) \cdot I_n\), where \(I_n\) is the identity matrix of the same order and \(\text{det}(A)\) is the determinant of \(A\). The adjugate matrix plays a crucial role in matrix inverse calculation.

Identify and Introduce the Given Problem Elements

We are given a matrix \(A\) and a scalar \(u\). We want to show that \(\text{adj}(uA) = u^{n-1} \cdot \text{adj}(A)\) for an \(n \times n\) matrix \(A\). Here, \(\text{adj}(uA)\) is the adjugate of the matrix \(uA\), where \(uA\) is obtained by multiplying each element of \(A\) by the scalar \(u\).

Derive the Determinant of a Scalar-Multiplied Matrix

For an \(n \times n\) matrix \(A\) where \(uA\) is the result of multiplying each element by \(u\), the determinant \(\text{det}(uA)\) is equal to \(u^n \cdot \text{det}(A)\). This property comes from the fact that multiplying an entire matrix by a scalar multiplies its determinant by that scalar raised to the power of the order of the matrix.

Relate the Adjugate of the Scaled Matrix to the Original Matrix

Now consider the definition \(A \cdot \text{adj}(A) = \text{det}(A) \cdot I_n\). Replace \(A\) by \(uA\) to obtain \((uA) \cdot \text{adj}(uA) = \text{det}(uA) \cdot I_n\). Substitute \(\text{det}(uA) = u^n \cdot \text{det}(A)\) to rewrite this as \((uA) \cdot \text{adj}(uA) = u^n \cdot \text{det}(A) \cdot I_n\).

Use Properties of Scalars and Matrices

Since \((uA) = u \cdot A\), rewrite the equation from Step 4 as \(u \cdot A \cdot \text{adj}(uA) = u^n \cdot \text{det}(A) \cdot I_n\). Divide both sides of the equation by \(u\) (assuming \(u eq 0\)) to get \(A \cdot \text{adj}(uA) = u^{n-1} \cdot \text{det}(A) \cdot I_n\).

Conclude the Argument

The result from Step 5, \(A \cdot \text{adj}(uA) = u^{n-1} \cdot \text{det}(A) \cdot I_n\), mirrors the adjugate definition: \(A \cdot \text{adj}(A) = \text{det}(A) \cdot I_n\). Thus, comparing these equations, we find that \(\text{adj}(uA) = u^{n-1} \cdot \text{adj}(A)\). This completes the proof.

Key concepts

Matrix Multiplication

When two matrices are multiplied together within linear algebra, the resulting product matrix is calculated by performing a series of operations. For two conformable matrices, say matrix \( A \) with dimensions \( m \times n \) and matrix \( B \) with dimensions \( n \times p \), the result \( AB \) will be a matrix of dimensions \( m \times p \). To find each element in the resulting matrix:

• Take the dot product of the corresponding row from the first matrix and column from the second matrix.
• Multiply each element of the row by the corresponding element of the column.
• Add all these products to get the element in the new matrix.

Matrix multiplication is essential in various applications including in the transformation of coordinates, finding powers of matrices, and solving systems of linear equations.

Determinant Properties

The determinant of a square matrix is a scalar value that provides essential information about the matrix. It can signal if a matrix is invertible among other properties. Importantly, determinants have several key properties:

• The determinant of a product of matrices is equal to the product of their determinants: \( \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \).
• For a matrix multiplied by a scalar \( u \), \( \text{det}(uA) = u^n \cdot \text{det}(A) \), where \( n \) is the order of the square matrix.
• If a matrix is multiplied by the identity matrix, its determinant remains the same.

Understanding these properties is crucial for simplifying complex matrix expressions, especially when calculating the inverse or adjugate of matrices.

Scalar Multiplication of Matrices

Scalar multiplication in matrices involves multiplying every entry in a matrix by the same scalar value. If you have a matrix \( A \) and a scalar \( u \), the resulting matrix \( uA \) is formed by:

• Multiplying each element of \( A \) by \( u \).

This operation maintains the size of the matrix but scales all its values. Scalar multiplication simplifies many operations with matrices, particularly when involved in determinant calculations. As shown earlier, multiplying a matrix by a scalar scales its determinant significantly depending on the matrix's size, specifically by the scalar raised to the power equal to the matrix's order.

Matrix Inverse Calculation

The inverse of a matrix, often denoted as \( A^{-1} \), is a matrix which, when multiplied with the original matrix \( A \), produces the identity matrix. However, not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible. The inverse is calculated using the cofactor matrix and the matrix's determinant. To find the inverse:

• Compute the adjugate (or adjoint) of the matrix by transposing its cofactor matrix.
• Divide the adjugate by the determinant of the original matrix.

Formally, if a matrix \( A \) is invertible, then \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \). This operation is powerful for solving linear equations and transforming matrices in numerous mathematical and real-world applications.