First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightMatrices And Determinantschevron_rightExercise 4.2

Solution

Let $A$ be a $3\times 3$ matrix.

Then $kA$ is obtained by multiplying every entry (equivalently every row) of $A$ by $k$ .

A basic determinant property is:

If one row of a determinant is multiplied by $k$ , the determinant is multiplied by $k$ .

Since $kA$ has each of the 3 rows multiplied by $k$ , we get:

|kA|=k\cdot k\cdot k\,|A|=k^3|A|

\boxed{|kA|=k^3|A|}