# Let two matrices A = $\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$ and B = $\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$ are two matrices such that they are commutative and c$ \ne $3b. Then, find the value of $\dfrac{{d - a}}{{3b - c}}$.

Answer

Verified

360.9k+ views

Hint: In order to solve this problem we will use the property commutative, since it is provided that the two matrices are commutative using this data and then equating the two obtained matrices after multiplication and solving to get the asked term you will reach the right answer.

Complete step-by-step answer:

We have A = $\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$and B = $\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$.

It's given that A and B are commutative.

It means AB = BA ……(1)

First we calculate AB.

So, AB = $\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$$\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$=$\left[ \begin{gathered}

{\text{a + 2c}}\,\,\,\,\,\,\,\,{\text{b + 2d}} \\

{\text{3a + 4c}}\,\,\,\,\,\,\,\,{\text{3b + 4d}} \\

\end{gathered} \right]$

Then we find BA.

So, BA =$\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$$\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$ = \[\left[ \begin{gathered}

{\text{a + 3b}}\,\,\,\,{\text{2a + 4b}} \\

{\text{c + 3d}}\,\,\,\,{\text{2c + 4d}} \\

\end{gathered} \right]\]

From (1) we can equate the value of AB and BA.

So, AB = BA

$\left[ \begin{gathered}

{\text{a + 2c}}\,\,\,\,\,\,\,\,{\text{b + 2d}} \\

{\text{3a + 4c}}\,\,\,\,\,\,\,\,{\text{3b + 4d}} \\

\end{gathered} \right]$=\[\left[ \begin{gathered}

{\text{a + 3b}}\,\,\,\,{\text{2a + 4b}} \\

{\text{c + 3d}}\,\,\,\,{\text{2c + 4d}} \\

\end{gathered} \right]\]

Now we can say, a+2c = a+3b

2c = 3b

So, c = $\dfrac{{{\text{3b}}}}{2}$ ……(2)

And also, b + 2d = 2a + 4b

2d – 2a = 3b

d - a = $\dfrac{{{\text{3b}}}}{{\text{2}}}$ ……(3)

Therefore, we can do $\dfrac{{{\text{d - a}}}}{{{\text{3b - c}}}}$= $\dfrac{{\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}{{{\text{3b - }}\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}$= $\dfrac{{\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}{{\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}$ = 1.

Hence, the value of $\dfrac{{{\text{d - a}}}}{{{\text{3b - c}}}}$= 1.

Note: Whenever you face such types of problems you have to use the properties of matrix. The properties used here is multiplication of matrices and addition of matrices. Then we have just solved the asked term by equating the matrix as it is given that the matrix is commutative. Doing this will give you the right answer.

Complete step-by-step answer:

We have A = $\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$and B = $\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$.

It's given that A and B are commutative.

It means AB = BA ……(1)

First we calculate AB.

So, AB = $\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$$\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$=$\left[ \begin{gathered}

{\text{a + 2c}}\,\,\,\,\,\,\,\,{\text{b + 2d}} \\

{\text{3a + 4c}}\,\,\,\,\,\,\,\,{\text{3b + 4d}} \\

\end{gathered} \right]$

Then we find BA.

So, BA =$\left[ \begin{gathered}

{\text{a}}\,\,\,\,{\text{b}} \\

{\text{c}}\,\,\,\,{\text{d}} \\

\end{gathered} \right]$$\left[ \begin{gathered}

1\,\,\,\,2 \\

3\,\,\,\,4 \\

\end{gathered} \right]$ = \[\left[ \begin{gathered}

{\text{a + 3b}}\,\,\,\,{\text{2a + 4b}} \\

{\text{c + 3d}}\,\,\,\,{\text{2c + 4d}} \\

\end{gathered} \right]\]

From (1) we can equate the value of AB and BA.

So, AB = BA

$\left[ \begin{gathered}

{\text{a + 2c}}\,\,\,\,\,\,\,\,{\text{b + 2d}} \\

{\text{3a + 4c}}\,\,\,\,\,\,\,\,{\text{3b + 4d}} \\

\end{gathered} \right]$=\[\left[ \begin{gathered}

{\text{a + 3b}}\,\,\,\,{\text{2a + 4b}} \\

{\text{c + 3d}}\,\,\,\,{\text{2c + 4d}} \\

\end{gathered} \right]\]

Now we can say, a+2c = a+3b

2c = 3b

So, c = $\dfrac{{{\text{3b}}}}{2}$ ……(2)

And also, b + 2d = 2a + 4b

2d – 2a = 3b

d - a = $\dfrac{{{\text{3b}}}}{{\text{2}}}$ ……(3)

Therefore, we can do $\dfrac{{{\text{d - a}}}}{{{\text{3b - c}}}}$= $\dfrac{{\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}{{{\text{3b - }}\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}$= $\dfrac{{\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}{{\dfrac{{\text{3}}}{{\text{2}}}{\text{b}}}}$ = 1.

Hence, the value of $\dfrac{{{\text{d - a}}}}{{{\text{3b - c}}}}$= 1.

Note: Whenever you face such types of problems you have to use the properties of matrix. The properties used here is multiplication of matrices and addition of matrices. Then we have just solved the asked term by equating the matrix as it is given that the matrix is commutative. Doing this will give you the right answer.

Last updated date: 24th Sep 2023

•

Total views: 360.9k

•

Views today: 3.60k

Recently Updated Pages

What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE