1. **State the problem:** We have matrices $$A=\begin{pmatrix}2 & x \\ 3 & 1 \end{pmatrix}, B=\begin{pmatrix}2 & 1 \\ 1 & 4 \end{pmatrix}, C=\begin{pmatrix}3x+2 & 7 \\ 7-x & 7 \end{pmatrix}$$ Find expression for matrix product $AB$ in terms of $x$. Determine $x$ given $B^T A^T = C$. Solve simultaneously for $I_1$, $I_2$, and $I_3$ using determinants given: $$2I_1 + 3I_2 - 4I_3 = 26$$ $$I_1 - 5I_2 - 3I_3 = -87$$ $$-7I_1 + 2I_2 + 6I_3 = 12$$ --- 2. **Calculate $AB$:** Multiply matrices using rules: $$AB = \begin{pmatrix}2 & x \\ 3 & 1\end{pmatrix} \begin{pmatrix}2 & 1 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix}2\times 2 + x \times 1 & 2 \times 1 + x \times 4 \\ 3 \times 2 + 1 \times 1 & 3 \times 1 + 1 \times 4 \end{pmatrix} = \begin{pmatrix}4+x & 2+4x \\ 7 & 7 \end{pmatrix}$$ --- 3. **Find $x$ given $B^T A^T = C$:** Transpose matrices: $$B^T = \begin{pmatrix}2 & 1 \\ 1 & 4 \end{pmatrix}^T = \begin{pmatrix}2 & 1 \\ 1 & 4 \end{pmatrix}$$ $$A^T = \begin{pmatrix}2 & x \\ 3 & 1 \end{pmatrix}^T = \begin{pmatrix}2 & 3 \\ x & 1 \end{pmatrix}$$ Multiply $B^T A^T$: $$B^T A^T = \begin{pmatrix}2 & 1 \\ 1 & 4 \end{pmatrix} \begin{pmatrix}2 & 3 \\ x & 1 \end{pmatrix} = \begin{pmatrix}2 \times 2 + 1 \times x & 2 \times 3 + 1 \times 1 \\ 1 \times 2 + 4 \times x & 1 \times 3 + 4 \times 1 \end{pmatrix} = \begin{pmatrix}4 + x & 7 \\ 2 + 4x & 7 \end{pmatrix}$$ Set equal to $C$: $$\begin{pmatrix}4 + x & 7 \\ 2 + 4x & 7 \end{pmatrix} = \begin{pmatrix}3x + 2 & 7 \\ 7 - x & 7 \end{pmatrix}$$ Equate elements: From (1,1): $$4 + x = 3x + 2$$ $$4 - 2 = 3x - x$$ $$2 = 2x \implies x = 1$$ From (2,1): $$2 + 4x = 7 - x$$ $$2 + 4(1) = 7 - 1$$ $$6 = 6$$ (checks out) --- 4. **Solve the linear system for $I_1, I_2, I_3$ using determinants:** System: $$\begin{cases} 2I_1 + 3I_2 - 4I_3 = 26 \\ I_1 - 5I_2 - 3I_3 = -87 \\ -7I_1 + 2I_2 + 6I_3 = 12 \end{cases}$$ Coefficient matrix $D$: $$D = \begin{pmatrix} 2 & 3 & -4 \\ 1 & -5 & -3 \\ -7 & 2 & 6 \end{pmatrix}$$ Calculate $|D|$: $$|D| = 2 \begin{vmatrix} -5 & -3 \\ 2 & 6 \end{vmatrix} - 3 \begin{vmatrix} 1 & -3 \\ -7 & 6 \end{vmatrix} - 4 \begin{vmatrix} 1 & -5 \\ -7 & 2 \end{vmatrix}$$ Calculate minors: $$\begin{vmatrix} -5 & -3 \\ 2 & 6 \end{vmatrix} = (-5)(6) - (-3)(2) = -30 + 6 = -24$$ $$\begin{vmatrix} 1 & -3 \\ -7 & 6 \end{vmatrix} = 1(6) - (-3)(-7) = 6 - 21 = -15$$ $$\begin{vmatrix} 1 & -5 \\ -7 & 2 \end{vmatrix} = 1(2) - (-5)(-7) = 2 - 35 = -33$$ So: $$|D| = 2(-24) - 3(-15) - 4(-33) = -48 + 45 + 132 = 129$$ --- Calculate determinants for $I_1, I_2, I_3$ replacements: $D_{I_1}$ replaces 1st column with constants: $$D_{I_1} = \begin{pmatrix} 26 & 3 & -4 \\ -87 & -5 & -3 \\ 12 & 2 & 6 \end{pmatrix}$$ $$|D_{I_1}| = 26 \begin{vmatrix} -5 & -3 \\ 2 & 6 \end{vmatrix} - 3 \begin{vmatrix} -87 & -3 \\ 12 & 6 \end{vmatrix} - 4 \begin{vmatrix} -87 & -5 \\ 12 & 2 \end{vmatrix}$$ Compute minors: $$\begin{vmatrix} -5 & -3 \\ 2 & 6 \end{vmatrix} = -24$$ (previously computed) $$\begin{vmatrix} -87 & -3 \\ 12 & 6 \end{vmatrix} = (-87)(6) - (-3)(12) = -522 + 36 = -486$$ $$\begin{vmatrix} -87 & -5 \\ 12 & 2 \end{vmatrix} = (-87)(2) - (-5)(12) = -174 + 60 = -114$$ Calculate: $$|D_{I_1}| = 26(-24) - 3(-486) - 4(-114) = -624 + 1458 + 456 = 1290$$ --- $D_{I_2}$ replaces 2nd column: $$D_{I_2} = \begin{pmatrix} 2 & 26 & -4 \\ 1 & -87 & -3 \\ -7 & 12 & 6 \end{pmatrix}$$ $$|D_{I_2}| = 2 \begin{vmatrix} -87 & -3 \\ 12 & 6 \end{vmatrix} - 26 \begin{vmatrix} 1 & -3 \\ -7 & 6 \end{vmatrix} - 4 \begin{vmatrix} 1 & -87 \\ -7 & 12 \end{vmatrix}$$ Compute minors: $$\begin{vmatrix} -87 & -3 \\ 12 & 6 \end{vmatrix} = -486$$ (above) $$\begin{vmatrix} 1 & -3 \\ -7 & 6 \end{vmatrix} = 1(6)-(-3)(-7)=6-21=-15$$ $$\begin{vmatrix} 1 & -87 \\ -7 & 12 \end{vmatrix} = 1(12) - (-87)(-7) = 12 - 609 = -597$$ Calculate: $$|D_{I_2}| = 2(-486) - 26(-15) - 4(-597) = -972 + 390 + 2388 = 1806$$ --- $D_{I_3}$ replaces 3rd column: $$D_{I_3} = \begin{pmatrix} 2 & 3 & 26 \\ 1 & -5 & -87 \\ -7 & 2 & 12 \end{pmatrix}$$ $$|D_{I_3}| = 2 \begin{vmatrix} -5 & -87 \\ 2 & 12 \end{vmatrix} - 3 \begin{vmatrix} 1 & -87 \\ -7 & 12 \end{vmatrix} + 26 \begin{vmatrix} 1 & -5 \\ -7 & 2 \end{vmatrix}$$ Compute minors: $$\begin{vmatrix} -5 & -87 \\ 2 & 12 \end{vmatrix} = (-5)(12) - (-87)(2) = -60 + 174 = 114$$ $$\begin{vmatrix} 1 & -87 \\ -7 & 12 \end{vmatrix} = 1(12) - (-87)(-7) = 12 - 609 = -597$$ $$\begin{vmatrix} 1 & -5 \\ -7 & 2 \end{vmatrix} = 1(2) - (-5)(-7) = 2 - 35 = -33$$ Calculate: $$|D_{I_3}| = 2(114) - 3(-597) + 26(-33) = 228 + 1791 - 858 = 1161$$ --- 5. **Calculate currents:** $$I_1 = \frac{|D_{I_1}|}{|D|} = \frac{1290}{129} = 10$$ $$I_2 = \frac{|D_{I_2}|}{|D|} = \frac{1806}{129} = 14$$ $$I_3 = \frac{|D_{I_3}|}{|D|} = \frac{1161}{129} = 9$$ --- **Final answers:** Matrix product: $$AB = \begin{pmatrix}4 + x & 2 + 4x \\ 7 & 7 \end{pmatrix}$$ Value of $x$: $$x = 1$$ Currents: $$I_1 = 10, \quad I_2 = 14, \quad I_3 = 9$$