Matrix P = ((3, 2-x), (x, x−2)) is singular. (a) Form an equation in x and show that it reduces to x2+x−6=0…

Question

Matrix P = ((3, 2-x), (x, x−2)) is singular. (a) Form an equation in x and show that it reduces to x2+x−6=0. (b) Solve the equation x2+x−6=0.

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29. Matrix $P = \begin{pmatrix} x & 2-x \\ 3 & x-2 \end{pmatrix}$ is singular.

a) Form an equation in $x$ and show that it reduces to $x^2 + x - 6 = 0$ .

Calculate the determinant of matrix P.
For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the determinant is $ad - bc$ .
$\det(P) = x(x-2) - (2-x)(3)$
Set the determinant to zero for a singular matrix.
$x(x-2) - 3(2-x) = 0$
Expand and simplify the equation.
$x^2 - 2x - (6 - 3x) = 0$ $x^2 - 2x - 6 + 3x = 0$ $x^2 + x - 6 = 0$
The equation is reduced to $\boxed{x^2 + x - 6 = 0}$ .

b) Solve the equation $x^2 + x - 6 = 0$ .
Factor the quadratic equation.
We need two numbers that multiply to -6 and add to 1. These numbers are 3 and -2.
$(x+3)(x-2) = 0$
Set each factor to zero and solve for $x$ .
$x+3 = 0 \quad \text{or} \quad x-2 = 0$ $x = -3 \quad \text{or} \quad x = 2$
The solutions are $x = \boxed{-3}$ or $x = \boxed{2}$ .

30. On a particular day the official exchange rate of USD (US dollars) to ZWL (ZIG) was 1 USD : 28 ZWL.

a) Convert 840 ZWL to US Dollar.
Use the given exchange rate.
1 USD = 28 ZWL
Divide the ZWL amount by the exchange rate to find the USD amount.
$\text{USD} = \frac{840 \text{ ZWL}}{28 \text{ ZWL/USD}}$ $\text{USD} = 30$
So, 840 ZWL is equal to $\boxed{30 \text{ US Dollars}}$ .

b) Convert 25 US Dollar to ZWL (ZIG).
Use the given exchange rate.
1 USD = 28 ZWL
Multiply the USD amount by the exchange rate to find the ZWL amount.
$\text{ZWL} = 25 \text{ USD} \times 28 \text{ ZWL/USD}$ $\text{ZWL} = 700$
So, 25 US Dollars is equal to $\boxed{700 \text{ ZWL}}$ .

Matrix P = ((3, 2-x), (x, x-2)) is singular. (a) Form an equation in x and show that it reduces to x2+x−6=0. (b) Solve the equation x2+x−6=0.