Question

If I is a unit matrix of order 10, then the determinant of I is equal to
A) 10
B) 1
C) \[\dfrac{1}{{10}}\]
D) 9

Answer 1

Hint: Here, the given matrix is of order 10 and is a unit matrix. As unity refers to 1, the determinant of the unit matrix in any order is 1, this can be shown for different orders.
Order of a matrix refers to its number of rows and columns.
In a unit matrix the all the members in diagonal are 1 and the rest are zero

Complete step-by-step answer:
Calculating determinant for unit matrix I of various orders:
I of order 1; [1] = 1
I of order 2 ();
$\left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right] = \left| {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right|$
$\left| {{I_2}} \right| = 1$
I of order 3 ();
$\left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right] = \left| {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right|$
$\left| {{I_3}} \right| = 1(1 \times 1 - 0)$
$\left| {{I_3}} \right| = 1$


I of order 4 ();
$\left[ {\begin{array}{*{20}{c}}
  1&0&0&0 \\
  0&1&0&0 \\
  0&0&1&0 \\
  0&0&0&1
\end{array}} \right] = \left| {\begin{array}{*{20}{c}}
  1&0&0&0 \\
  0&1&0&0 \\
  0&0&1&0 \\
  0&0&0&1
\end{array}} \right|$
$\left| {{I_4}} \right| = 1\left| {\begin{array}{*{20}{c}}
  1&0&0&0 \\
  0&1&0&0 \\
  0&0&1&0
\end{array}} \right|$
$\left| {{I_4}} \right| = 1\left| {{I_3}} \right|$
$\left| {{I_4}} \right| = 1$
Therefore, it can be seen that every order of unit matrix gives determinant 1.
Thus, determinant of unit matrix of order 10 is also 1 and it can be represented as:
\[\left[ {\begin{array}{*{20}{c}}
  1&0&0&0&0&0&0&0&0&0 \\
  0&1&0&0&0&0&0&0&0&0 \\
  1&0&1&0&0&0&0&0&0&0 \\
  1&0&0&1&0&0&0&0&0&0 \\
  1&0&0&0&1&0&0&0&0&0 \\
  1&0&0&0&0&1&0&0&0&0 \\
  1&0&0&0&0&0&1&0&0&0 \\
  1&0&0&0&0&0&0&1&0&0 \\
  1&0&0&0&0&0&0&0&1&0 \\
  1&0&0&0&0&0&0&0&0&1
\end{array}} \right] = \left| {\begin{array}{*{20}{c}}
  1&0&0&0&0&0&0&0&0&0 \\
  0&1&0&0&0&0&0&0&0&0 \\
  1&0&1&0&0&0&0&0&0&0 \\
  1&0&0&1&0&0&0&0&0&0 \\
  1&0&0&0&1&0&0&0&0&0 \\
  1&0&0&0&0&1&0&0&0&0 \\
  1&0&0&0&0&0&1&0&0&0 \\
  1&0&0&0&0&0&0&1&0&0 \\
  1&0&0&0&0&0&0&0&1&0 \\
  1&0&0&0&0&0&0&0&0&1
\end{array}} \right|\]
$\left| {{I_{10}}} \right| = 1$
So, the correct answer is “Option B”.

Note: Unit matrix is also called as Identity matrix
Order of matrix can also be represented as the product of number of rows and columns like (10 X 10)
Matrix is written inside brackets ‘[]’ where as the determinants is denoted by bars
Determinant for a 3 X 3 matrix can be calculated as:
$A = \left[ {\begin{array}{*{20}{c}}
  a&b&c \\
  d&e&f \\
  g&h&i
\end{array}} \right] = \left| {\begin{array}{*{20}{c}}
  a&b&c \\
  d&e&f \\
  g&h&i
\end{array}} \right|$
$|A| = {\text{ }}a(ei - fh) - b{\text{ }}(di - fg){\text{ }} + {\text{ }}c{\text{ }}(dh - eg)$