Question

If the kernel of a 5 × 4 matrix A consists of the zero-vector alone, and if AB = AC for two 4 × 5 matrices B and C, then matrices B and C must be equal.

Short answer

The above statement is true.

If the kernel of a 5 × 4 matrix A consists of the zero-vector alone, and if AB = AC for

two 4 × 5 matrices B and C, then matrices B and C must be equal.

Step-by-step solution

Definition of the kernel of a matrix

Let A be a matrix from ${\mathrm{ℝ}}^n\text{to}{\mathrm{ℝ}}^m$, then the kernel of the matrix A, denoted by ker(A), is defined as-

$ker(A)=\left\{x\in {\mathrm{ℝ}}^n:Ax=0\right\}$

To prove matrix B = C

If the kernel of a matrix A consists of zero vector only, then the equation

$Ax=0\Rightarrow x=0$

We have given,

$$\begin{gathered} AB=BC \\ \Rightarrow AB-BC=0 \\ \Rightarrow A\left(B-C\right)=0 \end{gathered}$$

Since it is given that the kernel of a 5 x 4 matrix A is zero-vector then

$$\begin{gathered} A\left(B-C\right)=0 \\ \Rightarrow B-C=0 \\ \Rightarrow B=C \end{gathered}$$

Final Answer

If the kernel of a 5 × 4 matrix A consists of the zero-vector alone, and if AB = AC for

two 4 × 5 matrices B and C, then matrices B and C must be equal.