Q.Find the values of $p, q, r,$ and $s$ if

[!FORMULA] $\begin{pmatrix} p^2-1 & 0 & -31-q^3 \\ 7 & r+1 & 9 \\ -2 & 8 & s-1\end{pmatrix} = \begin{pmatrix} 1 & 0 & -4 \\ 7 & \frac32 & 9 \\ -2 & 8 & -\pi\end{pmatrix}$

By Definition 7.13, $A=B$ (same order) forces $a_{ij}=b_{ij}$ for every $i,j$. Comparing the entries that carry the unknowns:

Step 1. Compare the $(1,1)$ entries. $p^2-1=1 \Rightarrow p^2=2 \Rightarrow p=\pm\sqrt2$.

Step 2. Compare the $(1,3)$ entries. $-31-q^3=-4 \Rightarrow -q^3=-4+31=27 \Rightarrow q^3=-27=(-3)^3 \Rightarrow q=-3$.

Step 3. Compare the $(2,2)$ entries. $r+1=\dfrac32 \Rightarrow r=\dfrac32-1=\dfrac12$.

Step 4. Compare the $(3,3)$ entries. $s-1=-\pi \Rightarrow s=1-\pi$.

All the remaining entries ($7,0,9,-2,8$) already agree on both sides, so they impose no further condition.

• Forgetting that $p^2=2$ gives TWO values, $p=\pm\sqrt2$
• Sign slip while isolating $q^3$ from $-31-q^3=-4$