A is matrix (5 3 / 1 7) and I is the 2 × 2 identity matrix. Determine the values of h and k for which / = hI…

Question

A is matrix (5 3 / 1 7) and I is the 2 × 2 identity matrix. Determine the values of h and k for which / = hI + kA.

ScanSolve AI Tutor · Accepted Answer

Haya Chacha, hapa kuna suluhisho la maswali yako:

15.
Ili kukokotoa asilimia ya hitilafu, tunatumia fomula:

\text{Asilimia ya Hitilafu} = \frac{|\text{Thamani Iliyopimwa} - \text{Thamani Halisi}|}{\text{Thamani Halisi}} \times 100\%

Tambua thamani halisi na thamani iliyopimwa.
Thamani Halisi (kipenyo halisi) $= 5 \text{ mm}$
Thamani Iliyopimwa (kipenyo kilichorekodiwa) $= 4.986 \text{ mm}$
Kokotoa tofauti kamili (absolute error).
$\text{Tofauti kamili} = |4.986 \text{ mm} - 5 \text{ mm}|$ $\text{Tofauti kamili} = |-0.014 \text{ mm}|$ $\text{Tofauti kamili} = 0.014 \text{ mm}$
Kokotoa asilimia ya hitilafu.
$\text{Asilimia ya Hitilafu} = \frac{0.014 \text{ mm}}{5 \text{ mm}} \times 100\%$ $\text{Asilafu ya Hitilafu} = 0.0028 \times 100\%$ $\text{Asilimia ya Hitilafu} = 0.28\%$
Asilimia ya hitilafu katika kipimo hiki ni \boxed{\text{0.28%}}.

16.
Tunapewa matrix $A = \begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix}$ na $I$ ni matrix ya utambulisho ya $2 \times 2$ , ambayo ni $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ . Tunahitaji kupata thamani za $h$ na $k$ kwa mlinganyo $A = hI + kA$ .
Badilisha matrices $A$ na $I$ kwenye mlinganyo.
$\begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix} = h \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + k \begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix}$
Fanya kuzidisha kwa scalar kwa $hI$ na $kA$ .
$\begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix} = \begin{pmatrix} h \cdot 1 & h \cdot 0 \\ h \cdot 0 & h \cdot 1 \end{pmatrix} + \begin{pmatrix} k \cdot 5 & k \cdot 3 \\ k \cdot 1 & k \cdot 7 \end{pmatrix}$ $\begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix} = \begin{pmatrix} h & 0 \\ 0 & h \end{pmatrix} + \begin{pmatrix} 5k & 3k \\ k & 7k \end{pmatrix}$
Ongeza matrices upande wa kulia.
$\begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix} = \begin{pmatrix} h + 5k & 0 + 3k \\ 0 + k & h + 7k \end{pmatrix}$ $\begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix} = \begin{pmatrix} h + 5k & 3k \\ k & h + 7k \end{pmatrix}$
Linganisha vipengele vinavyolingana vya matrices ili kuunda milinganyo.
Kutoka kwa kipengele cha juu kulia:
$3 = 3k$
Kutoka kwa kipengele cha chini kushoto:
$1 = k$
Kutoka kwa milinganyo hii miwili, tunaweza kuona kwamba $k=1$ .
Badilisha $k=1$ kwenye mlinganyo wowote unaohusisha $h$ na $k$ .
Tutatumia kipengele cha juu kushoto:
$5 = h + 5k$ $5 = h + 5(1)$ $5 = h + 5$ $h = 5 - 5$ $h = 0$
Thibitisha kwa kutumia kipengele cha chini kulia
$7 = h + 7k$ $7 = 0 + 7(1)$ $7 = 7$
Hii inathibitisha thamani za $h$ na $k$ .

Thamani za $h$ na $k$ ni $\boxed{h=0, k=1}$ .

A is matrix (5 3 / 1 7) and I is the 2 x 2 identity matrix. Determine the values of h and k for which / = hI + kA.

Answer

Step-by-step solution

Tambua thamani halisi na thamani iliyopimwa.

Kokotoa tofauti kamili (absolute error).

Kokotoa asilimia ya hitilafu.

Badilisha matrices $A$ na $I$ kwenye mlinganyo.

Fanya kuzidisha kwa scalar kwa $hI$ na $kA$ .

Ongeza matrices upande wa kulia.

Linganisha vipengele vinavyolingana vya matrices ili kuunda milinganyo.

Badilisha $k=1$ kwenye mlinganyo wowote unaohusisha $h$ na $k$ .

Thibitisha kwa kutumia kipengele cha chini kulia

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