Question

Prove that if $A$ is a $n \times n$ matrix and $c$ is a scalar, then $\operatorname{det}(c A)=c^n \operatorname{det} A$.

   Prove that if $A$ is a $n \times n$ matrix and $c$ is a scalar, then $\operatorname{det}(c A)=c^n \operatorname{det} A$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 1, Problem 6

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Step 1

We need to prove that for any scalar \( c \) and any \( n \times n \) matrix \( A \), the determinant of the matrix \( cA \) (where every element of \( A \) is multiplied by \( c \)) is equal to \( c^n \) times the determinant of \( A \). Show more…

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Prove that if $A$ is a $n \times n$ matrix and $c$ is a scalar, then $\operatorname{det}(c A)=c^n \operatorname{det} A$.

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Key Concepts

Determinant

The determinant is a scalar value assigned to a square matrix that captures key properties of the matrix, such as invertibility and volume scaling in geometric transformations. It is computed from the matrix entries using a specific formula, such as cofactor expansion or permutation summation, and is fundamental in linear algebra.

Multilinearity of the Determinant

Multilinearity means that the determinant is a linear function in each row (or column) when the others are held fixed. This property ensures that if one row of the matrix is multiplied by a scalar, the entire determinant is multiplied by that scalar. It is a key concept for proving how scaling rows of a matrix affects the overall determinant.

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying each entry of the matrix by a given scalar. In terms of determinants, this process effectively multiplies each row (or column) by the scalar. Because the determinant is multilinear, if each of the n rows is multiplied by the scalar, the determinant is scaled by the scalar raised to the power n, which is essential to proving the formula det(cA) = c^n det(A).

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