Finding Missing Values in a Matrix

Worked example: using a stated property

If $\begin{bmatrix} x+2 & y-3 \\ 0 & 4 \end{bmatrix}$ is a scalar matrix, find $x$ and $y$ .

Solution: A scalar matrix needs every off-diagonal entry to be zero and both diagonal entries equal. The off-diagonal entry $y-3$ must be zero:

y - 3 = 0 \;\Rightarrow\; y = 3

And the diagonal entries must match: $x+2 = 4$ , so $x=2$ .

If $\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$ , find $x$ .

Solution: Evaluate both determinants:

3(1) - x(x) = 3(1) - 2(4) \;\Rightarrow\; 3-x^2 = -5

x^2 = 8 \;\Rightarrow\; x = \pm 2\sqrt{2}

Finding Missing Values in a Matrix

If $\begin{bmatrix} x+2 & y-3 \\ 0 & 4 \end{bmatrix}$ is a scalar matrix, find $x$ and $y$ .

Solution: A scalar matrix needs every off-diagonal entry to be zero and both diagonal entries equal. The off-diagonal entry $y-3$ must be zero:

y - 3 = 0 \;\Rightarrow\; y = 3

And the diagonal entries must match: $x+2 = 4$ , so $x=2$ .

If $\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$ , find $x$ .

Solution: Evaluate both determinants:

3(1) - x(x) = 3(1) - 2(4) \;\Rightarrow\; 3-x^2 = -5

x^2 = 8 \;\Rightarrow\; x = \pm 2\sqrt{2}