Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES4 Marks
Question
If AB = BA for any two sqaure matrices, prove by mathematical induction that (AB)n = AnBn.
✓

Answer

Let P(n) : (AB)n = AnBn

$\therefore$ P(1) : (AB)1 = A1B1 ⇒ AB = AB

So, P(1) is true.

Now, P(k) : (AB)k = AkBk $\text{k}\in\text{N}$

So, P(k) is true, whenever P(k + 1) is true.

$\therefore$ P(k + 1) ⇒ (AB)k+1 = Ak+1.Bk+1

$\therefore$ P(k + 1 : AB)k+1 = Ak+1Bk+1

⇒ Ak.Bk.AB

⇒ Ak.BkBA ⇒ AkBk+1A

⇒ AkA.Bk+1 ⇒ Ak+1Bk+1

⇒ (A.B)k+1 = Ak+1Bk+1

So, P(k+1) is true for all $\text{n}\in\text{N},$ whenever P(k) is true.

By mathematical induction (AB) = AnBn is true for all $\text{n}\in\text{N}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from MATRICES→MATRICES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If f : R → (-1, 1) defined by $\text{f(x)}=\frac{10^{\text{x}}-10^{-\text{x}}}{10^{\text{x}}+10^{-\text{x}}}$ is invertible, find f-1.
→
Solve the following equation for x:
$\tan^{-1}2\text{x}+\tan^{-1}3\text{x}=\text{n}\pi+\frac{3\pi}{4}$
→
Write the minimum value of f(x) = xx .
→
Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$
→
Find the shortest distance between the lines
$\vec{\text{r}}=\Big(4\hat{\text{i}}-\hat{\text{j}}\Big)+\lambda\Big(\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\Big)$ $\text{and}\ \vec{\text{r}}=\Big(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}\Big)+\mu\Big(2\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}\Big).$
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\sin\text{x}\sin2\text{x}\sin3\text{x}\sin4\text{x}$
→
If $\text{A}^{-1}=\begin{bmatrix}3 & -1 & 1 \\ -15 & 5 & -5 \\ 5 & -2 & 2\end{bmatrix}$and

$\text{And B}=\begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix},$ find (AB)-1.

→
If $(\cos\text{x})^\text{y}=(\cos\text{y})^\text{x},$ find $\frac{\text{dy}}{\text{dx}}$
→
Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as:
$\text{a}\times\text{b}=\begin{cases}\text{a + b},&\text{if }\text{a + b}<6\\\text{a + b}-6,&\text{if }\text{a + b}\geq6\end{cases}$
Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
→
To maintain one's health, a person must fulfil certain minimum daily requirements for the following three nutrients: calcium, protein and calories. The diet consists of only items I and II whose prices and nutrient contents are shown below:
  Food I Food II Minimum daily requirement
Calcium 10 4 20
Protein 5 6 20
Calories 2 6 12
Price Rs. 0.60 per unit Rs. 1.00 per unit  
Find the combination of food items so that the cost may be minimum.
→