Answer

Verified

450k+ views

Hint: For any general 2 x 2 matrix, \[M=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\], \[{{M}^{2}}=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\times \left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]=\left[ \begin{matrix}

{{a}^{2}}+bc & ab+bd \\

ca+dc & cb+{{d}^{2}} \\

\end{matrix} \right]\]. So, to find \[{{A}^{2}}\], use this formula of \[{{M}^{2}}\] by considering a = i, b = 0, c = 0 and d = i.

We are given a 2 x 2 matrix, \[A=\left[ \begin{matrix}

i & 0 \\

0 & i \\

\end{matrix} \right]\]

Here, we have to find \[{{A}^{2}}\].

Let us take the matrix given in the question,

\[A=\left[ \begin{matrix}

i & 0 \\

0 & i \\

\end{matrix} \right]\] which is a 2 x 2 matrix as it has 2 rows and 2 columns.

Let us taken the general 2 x 2 matrix, that is

\[M=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\]

Now, we know that \[{{M}^{2}}=M\times M....\left( i \right)\]

By, putting the general 2 x 2 matrix in place of M in the right hand side (RHS) of the equation (i), we get,

\[{{M}^{2}}={{\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]}_{2\times 2}}\times {{\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]}_{2\times 2}}\]

We know that matrix multiplication is carried out by multiplying rows of the first matrix to columns of the second matrix. Therefore, we get above expression as,

\[{{M}^{2}}=\left[ \begin{matrix}

a\times a+b\times c & a\times b+b\times d \\

c\times a+d\times c & c\times b+d\times d \\

\end{matrix} \right].....\left( ii \right)\]

By simplifying the above expression, we get,

\[{{M}^{2}}=\left[ \begin{matrix}

{{a}^{2}}+bc & ab+bd \\

ca+dc & cb+{{d}^{2}} \\

\end{matrix} \right]\]

Now, we will compare the matrix given in the question, that is \[A=\left[ \begin{matrix}

i & 0 \\

0 & i \\

\end{matrix} \right]\] with general 2 x 2 matrix \[M=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\] , so we will get,

\[\begin{align}

& a=i \\

& b=0 \\

& c=0 \\

& d=i \\

\end{align}\]

Now, to get the value of \[{{A}^{2}}\], we will put these values of a, b, c and d in equation (ii). So we will get,

\[{{A}^{2}}=\left[ \begin{matrix}

{{\left( i \right)}^{2}}+0\times 0 & i\times \left( 0 \right)+\left( 0 \right)\times i \\

0\times i+i\times 0 & 0\times 0+{{\left( i \right)}^{2}} \\

\end{matrix} \right]\]

By simplifying the above expression, we get,

\[{{A}^{2}}=\left[ \begin{matrix}

{{i}^{2}} & 0 \\

0 & {{i}^{2}} \\

\end{matrix} \right]\]

As we know that it is an imaginary number and its value is \[\sqrt{-1}\]. Therefore, we get \[{{i}^{2}}=-1\].

Hence, we finally get \[{{A}^{2}}\] as \[\left[ \begin{matrix}

{{i}^{2}} & 0 \\

0 & {{i}^{2}} \\

\end{matrix} \right]=\left[ \begin{matrix}

-1 & 0 \\

0 & -1 \\

\end{matrix} \right]\].

Note: Students must note that to perform the matrix multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix. Students should also remember that matrix multiplication is only carried out by multiplying rows of the first matrix by columns of the second matrix unlike in determinant. In determinant, multiplication can be carried out by multiplying row to row and column to column as well. Therefore, students must not confuse between multiplication of two matrices or two determinants. Also, take special care in taking the values of the variables a, b, c and d.

a & b \\

c & d \\

\end{matrix} \right]\], \[{{M}^{2}}=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\times \left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]=\left[ \begin{matrix}

{{a}^{2}}+bc & ab+bd \\

ca+dc & cb+{{d}^{2}} \\

\end{matrix} \right]\]. So, to find \[{{A}^{2}}\], use this formula of \[{{M}^{2}}\] by considering a = i, b = 0, c = 0 and d = i.

We are given a 2 x 2 matrix, \[A=\left[ \begin{matrix}

i & 0 \\

0 & i \\

\end{matrix} \right]\]

Here, we have to find \[{{A}^{2}}\].

Let us take the matrix given in the question,

\[A=\left[ \begin{matrix}

i & 0 \\

0 & i \\

\end{matrix} \right]\] which is a 2 x 2 matrix as it has 2 rows and 2 columns.

Let us taken the general 2 x 2 matrix, that is

\[M=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\]

Now, we know that \[{{M}^{2}}=M\times M....\left( i \right)\]

By, putting the general 2 x 2 matrix in place of M in the right hand side (RHS) of the equation (i), we get,

\[{{M}^{2}}={{\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]}_{2\times 2}}\times {{\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]}_{2\times 2}}\]

We know that matrix multiplication is carried out by multiplying rows of the first matrix to columns of the second matrix. Therefore, we get above expression as,

\[{{M}^{2}}=\left[ \begin{matrix}

a\times a+b\times c & a\times b+b\times d \\

c\times a+d\times c & c\times b+d\times d \\

\end{matrix} \right].....\left( ii \right)\]

By simplifying the above expression, we get,

\[{{M}^{2}}=\left[ \begin{matrix}

{{a}^{2}}+bc & ab+bd \\

ca+dc & cb+{{d}^{2}} \\

\end{matrix} \right]\]

Now, we will compare the matrix given in the question, that is \[A=\left[ \begin{matrix}

i & 0 \\

0 & i \\

\end{matrix} \right]\] with general 2 x 2 matrix \[M=\left[ \begin{matrix}

a & b \\

c & d \\

\end{matrix} \right]\] , so we will get,

\[\begin{align}

& a=i \\

& b=0 \\

& c=0 \\

& d=i \\

\end{align}\]

Now, to get the value of \[{{A}^{2}}\], we will put these values of a, b, c and d in equation (ii). So we will get,

\[{{A}^{2}}=\left[ \begin{matrix}

{{\left( i \right)}^{2}}+0\times 0 & i\times \left( 0 \right)+\left( 0 \right)\times i \\

0\times i+i\times 0 & 0\times 0+{{\left( i \right)}^{2}} \\

\end{matrix} \right]\]

By simplifying the above expression, we get,

\[{{A}^{2}}=\left[ \begin{matrix}

{{i}^{2}} & 0 \\

0 & {{i}^{2}} \\

\end{matrix} \right]\]

As we know that it is an imaginary number and its value is \[\sqrt{-1}\]. Therefore, we get \[{{i}^{2}}=-1\].

Hence, we finally get \[{{A}^{2}}\] as \[\left[ \begin{matrix}

{{i}^{2}} & 0 \\

0 & {{i}^{2}} \\

\end{matrix} \right]=\left[ \begin{matrix}

-1 & 0 \\

0 & -1 \\

\end{matrix} \right]\].

Note: Students must note that to perform the matrix multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix. Students should also remember that matrix multiplication is only carried out by multiplying rows of the first matrix by columns of the second matrix unlike in determinant. In determinant, multiplication can be carried out by multiplying row to row and column to column as well. Therefore, students must not confuse between multiplication of two matrices or two determinants. Also, take special care in taking the values of the variables a, b, c and d.

Recently Updated Pages

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Let x1x2xn be in an AP of x1 + x4 + x9 + x11 + x20-class-11-maths-CBSE

Let x1x2x3 and x4 be four nonzero real numbers satisfying class 11 maths CBSE

Trending doubts

How many crores make 10 million class 7 maths CBSE

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples of Material nouns Abstract nouns Common class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

List out three methods of soil conservation

Write an application to the principal requesting five class 10 english CBSE