0000:00:00:00:00Start14Number 14ExplanationGiven matrices A=(1213)A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}A=(1123) and B=(4113)B = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix}B=(4113) Matrix CCC of order 2×22 \times 22×2 satisfies AC=BAC = BAC=B, the determinant of matrix CCC is ....111666999111111121212ExplanationGiven A=(1213)A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}A=(1123) and B=(4113)B = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix}B=(4113) with AC=BAC = BAC=B Finding Matrix C From the equation AC=BAC = BAC=B, multiply both sides by A−1A^{-1}A−1 AC=BAC = BAC=BA−1AC=A−1BA^{-1}AC = A^{-1}BA−1AC=A−1BC=A−1BC = A^{-1}BC=A−1B However, since the order of matrix multiplication matters C=BA−1C = BA^{-1}C=BA−1 Determining Inverse of Matrix A For matrix A=(1213)A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}A=(1123), its inverse is A−1=1(1⋅3)−(1⋅2)(3−2−11)A^{-1} = \frac{1}{(1 \cdot 3) - (1 \cdot 2)} \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}A−1=(1⋅3)−(1⋅2)1(3−1−21)=13−2(3−2−11)= \frac{1}{3 - 2} \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}=3−21(3−1−21)=(3−2−11)= \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}=(3−1−21) Calculating Matrix C Calculate C=BA−1C = BA^{-1}C=BA−1 C=(4113)(3−2−11)C = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}C=(4113)(3−1−21)=(4(3)+1(−1)4(−2)+1(1)1(3)+3(−1)1(−2)+3(1))= \begin{pmatrix} 4(3) + 1(-1) & 4(-2) + 1(1) \\ 1(3) + 3(-1) & 1(-2) + 3(1) \end{pmatrix}=(4(3)+1(−1)1(3)+3(−1)4(−2)+1(1)1(−2)+3(1))=(12−1−8+13−3−2+3)= \begin{pmatrix} 12 - 1 & -8 + 1 \\ 3 - 3 & -2 + 3 \end{pmatrix}=(12−13−3−8+1−2+3)=(11−701)= \begin{pmatrix} 11 & -7 \\ 0 & 1 \end{pmatrix}=(110−71) Calculating Determinant of C The determinant of matrix CCC is ∣C∣=(11⋅1)−(−7⋅0)|C| = (11 \cdot 1) - (-7 \cdot 0)∣C∣=(11⋅1)−(−7⋅0)=11−0=11= 11 - 0 = 11=11−0=11 Therefore, the determinant of matrix CCC is 111111
14Number 14ExplanationGiven matrices A=(1213)A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}A=(1123) and B=(4113)B = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix}B=(4113) Matrix CCC of order 2×22 \times 22×2 satisfies AC=BAC = BAC=B, the determinant of matrix CCC is ....111666999111111121212ExplanationGiven A=(1213)A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}A=(1123) and B=(4113)B = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix}B=(4113) with AC=BAC = BAC=B Finding Matrix C From the equation AC=BAC = BAC=B, multiply both sides by A−1A^{-1}A−1 AC=BAC = BAC=BA−1AC=A−1BA^{-1}AC = A^{-1}BA−1AC=A−1BC=A−1BC = A^{-1}BC=A−1B However, since the order of matrix multiplication matters C=BA−1C = BA^{-1}C=BA−1 Determining Inverse of Matrix A For matrix A=(1213)A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}A=(1123), its inverse is A−1=1(1⋅3)−(1⋅2)(3−2−11)A^{-1} = \frac{1}{(1 \cdot 3) - (1 \cdot 2)} \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}A−1=(1⋅3)−(1⋅2)1(3−1−21)=13−2(3−2−11)= \frac{1}{3 - 2} \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}=3−21(3−1−21)=(3−2−11)= \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}=(3−1−21) Calculating Matrix C Calculate C=BA−1C = BA^{-1}C=BA−1 C=(4113)(3−2−11)C = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}C=(4113)(3−1−21)=(4(3)+1(−1)4(−2)+1(1)1(3)+3(−1)1(−2)+3(1))= \begin{pmatrix} 4(3) + 1(-1) & 4(-2) + 1(1) \\ 1(3) + 3(-1) & 1(-2) + 3(1) \end{pmatrix}=(4(3)+1(−1)1(3)+3(−1)4(−2)+1(1)1(−2)+3(1))=(12−1−8+13−3−2+3)= \begin{pmatrix} 12 - 1 & -8 + 1 \\ 3 - 3 & -2 + 3 \end{pmatrix}=(12−13−3−8+1−2+3)=(11−701)= \begin{pmatrix} 11 & -7 \\ 0 & 1 \end{pmatrix}=(110−71) Calculating Determinant of C The determinant of matrix CCC is ∣C∣=(11⋅1)−(−7⋅0)|C| = (11 \cdot 1) - (-7 \cdot 0)∣C∣=(11⋅1)−(−7⋅0)=11−0=11= 11 - 0 = 11=11−0=11 Therefore, the determinant of matrix CCC is 111111
