a) A matrix A is singular. Find x and state the matrix A. b) Two shopkeepers, Juma and Wanjiku bought some items from a wholesale... The number of items bought by Juma and Wanjiku using a 2x3 matrix. The prices of the items bought using 3x1 matrix.

Hapa kuna suluhisho la swali lako:

20. a) Matrix A ni singular. Tafuta $x$ na ueleze matrix A.

Matrix A imetolewa kama: $A = \begin{bmatrix} x & x-1 \\ 2 & 4 \end{bmatrix}$

Matrix ni singular ikiwa determinant yake ni sifuri. Determinant ya matrix $2 \times 2$, $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, ni $ad - bc$.

Step 1: Kokotoa determinant ya matrix A. $\det(A) = (x)(4) - (x-1)(2)$ $\det(A) = 4x - (2x - 2)$ $\det(A) = 4x - 2x + 2$ $\det(A) = 2x + 2$

Step 2: Weka determinant sawa na sifuri na utatue kwa $x$. $2x + 2 = 0$ $2x = -2$ $x = \frac{-2}{2}$ $x = -1$

Step 3: Eleza matrix A kwa kutumia thamani ya $x$ uliyopata. Badilisha $x = -1$ kwenye matrix A: $A = \begin{bmatrix} -1 & (-1)-1 \\ 2 & 4 \end{bmatrix}$ A = \begin{bmatrix -1 & -2 \\ 2 & 4 \end{bmatrix}}

b) Wafanyabiashara wawili, Juma na Wanjiku walinunua bidhaa kutoka kwa muuzaji wa jumla.

i) Idadi ya bidhaa zilizonunuliwa na Juma na Wanjiku kwa kutumia matrix ya $2 \times 3$.

Juma alinunua mikate 18, pakiti 40 za maziwa, na sabuni 5. Wanjiku alinunua mikate 15, pakiti 30 za maziwa, na sabuni 6.

Matrix ya $2 \times 3$ (safu 2, nguzo 3) itawakilisha wanunuzi (Juma, Wanjiku) kama safu na bidhaa (mikate, maziwa, sabuni) kama nguzo.

\text{Idadi ya bidhaa} = \begin{bmatrix 18 & 40 & 5 \\ 15 & 30 & 6 \end{bmatrix}}

ii) Bei za bidhaa zilizonunuliwa kwa kutumia matrix ya $3 \times 1$.

Bei ya mkate ni Ksh. 45. Bei ya pakiti ya maziwa ni Ksh. 50. Bei ya sabuni ni Ksh. 150.

Matrix ya $3 \times 1$ (safu 3, nguzo 1) itawakilisha bei za bidhaa.

\text{Bei za bidhaa} = \begin{bmatrix 45 \\ 50 \\ 150 \end{bmatrix}}

iii) Jumla ya gharama iliyolipwa na kila mfanyabiashara.

Ili kupata jumla ya gharama iliyolipwa na kila mfanyabiashara, tunazidisha matrix ya idadi ya bidhaa na matrix ya bei.

Step 1: Zidisha matrix ya idadi ya bidhaa na matrix ya bei. $\text{Gharama} = \begin{bmatrix} 18 & 40 & 5 \\ 15 & 30 & 6 \end{bmatrix} \begin{bmatrix} 45 \\ 50 \\ 150 \end{bmatrix}$

Step 2: Kokotoa bidhaa ya matrix. Kwa Juma (safu ya kwanza): $(18 \times 45) + (40 \times 50) + (5 \times 150)$ $= 810 + 2000 + 750$ $= 3560$

Kwa Wanjiku (safu ya pili): $(15 \times 45) + (30 \times 50) + (6 \times 150)$ $= 675 + 1500 + 900$ $= 3075$

Step 3: Andika matokeo kama matrix. \text{Gharama} = \begin{bmatrix 3560 \\ 3075 \end{bmatrix}} Hivyo, Juma alilipa Ksh. 3560 na Wanjiku alilipa Ksh. 3075.

That's 4 down, 1 left today. What's next?