Chapter 6, Problem 6.1MCQ

For a set of linear algebraic equations in matrix format, $Ax-y,$ for a unique solution to exist, det(A) should be __________.

To determine

To fill in the blank space: For a set of linear algebraic equations in matrix format, $Ax=y$ , for a unique solution to exist, $\det \left(A\right)$ should be ________.

Answer to Problem 6.1MCQ

For a set of linear algebraic equations in matrix format, $Ax=y$ , for a unique solution to exist, $\det \left(A\right)$ should be non-zero.

Explanation of Solution

Given:

A set of linear algebraic equations in matrix format, $Ax=y$ .

If $\det \left(A\right)\ne 0$ , it follows that $A^{-1}=\frac{adj\left(A\right)}{\det \left(A\right)}$ is defined and exists.

Multiplying $A^{-1}$ to the left of both sides of the matrix equation $Ax=y$ , we get,

$\begin{array}{l}{A}^{-1}Ax=A^{-1}y\\ Ix=A^{-1}y\\ x=A^{-1}y\end{array}$

This implies that the solution of the given matrix equation is $x=A^{-1}y$ .

For a given matrix $A$ , $A^{-1}$ is uniquely defined.

Then, for given matrices $A$ and $y$ , $x=A^{-1}y$ is unique.

Thus, if $\det \left(A\right)\ne 0$ , $Ax=y$ has a unique solution.

So, filling in the blank space in the given statement:

For a set of linear algebraic equations in matrix format, $Ax=y$ , for a unique solution to exist, $\det \left(A\right)$ should be non-zero.