Animated Solution for Mathematics - Matrices and Determinants: For which of the following ordered pairs $(\mu ,\delta )$, the system of linear equations $x+2y+3z=1$, $3x+4y+5z=\mu$, $4x+4y+4z=\delta$ is inconsistent?

We are presented with a system of three linear equations representing planes in three-dimensional space:

Our objective is to identify the pair $(\mu ,\delta )$ that renders this system inconsistent, meaning the planes do not share a common point of intersection.

The Determinant

For a system to be inconsistent, the determinant of the coefficient matrix $D$ must be zero. Let us construct and evaluate this determinant:

Expanding along the first row:

Since $D=0$, the system is either inconsistent or possesses infinitely many solutions.

Row Dependence

To determine the nature of the system, we examine the relationship between the rows of the coefficient matrix. Let us test a linear combination of the first two rows:

This result is exactly the third row of our matrix. This confirms that the third row is linearly dependent on the first two, implying the third plane is geometrically constrained by the others.

The Condition for Inconsistency

For the system to be consistent, the constants on the right-hand side must satisfy the same linear relationship derived from the rows:

Conversely, for the system to be inconsistent, this pattern must be broken. Therefore, the condition for inconsistency is:

We evaluate the potential pairs $(\mu ,\delta )$ against the consistency condition $\delta =2\mu -2$:

1. For $(4,6)$: $2(4)-2=6$. Since $6=6$, the system is consistent. 2. For $(3,4)$: $2(3)-2=4$. Since $4=4$, the system is consistent. 3. For $(1,0)$: $2(1)-2=0$. Since $0=0$, the system is consistent. 4. For $(4,3)$: $2(4)-2=6$. Since $6 eq 3$, the pattern is broken.

The pair $(4,3)$ makes the system inconsistent.