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:

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: $A = \left[ {\begin{array}{*{20}{c}} a&b;&c; \\\ d&e;&f; \\\ g&h;&i; \end{array}} \right] = \left| {\begin{array}{*{20}{c}} a&b;&c; \\\ d&e;&f; \\\ g&h;&i; \end{array}} \right|$ $|A| = {\text{ }}a(ei - fh) - b{\text{ }}(di - fg){\text{ }} + {\text{ }}c{\text{ }}(dh - eg)$