Matrices
NCERT Solution
Exercise 2 Part 1
Question 1: Let A, B and C as shown below.
\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix} |
\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix} |
\begin{bmatrix}-2 & 5\\3 & 4 \end{bmatrix} |
Find each of the following:
- A + B
- A – B
- 3A – C
- AB
- BA
Solution:
(i)` A + B = ` |
\begin{bmatrix}2+1 & 4+3\\3+(-2) & 2+5\end{bmatrix} |
Or, `A + B =` |
\begin{bmatrix}3 & 7\\1 & 7\end{bmatrix} |
(ii)` A - B = ` |
\begin{bmatrix}2-1 & 4-3\\3-(-2) & 2-5\end{bmatrix} |
Or, `A + B =` |
\begin{bmatrix}1 & 1\\5 & -3\end{bmatrix} |
(iii) `3A-C=3` |
\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix} |
- |
\begin{bmatrix}-2 & 5\\3 & 4\end{bmatrix} |
= |
\begin{bmatrix}6 & 12\\9 & 6\end{bmatrix} |
- |
\begin{bmatrix}-2 & 5\\3 & 4\end{bmatrix} |
= |
\begin{bmatrix}6-(-2) & 12-5\\9-3 & 6-4\end{bmatrix} |
Or, `3A-B=` |
\begin{bmatrix}8 & 7\\6 & 2\end{bmatrix} |
(iv) `AB=` |
\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix} |
\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix} |
= |
\begin{bmatrix}2×1+4(-1) & 2 ×3+4×5\\3×1+2(-2) & 3× +2×5\end{bmatrix} |
= |
\begin{bmatrix}2-8 & 6+20\\3-4 & 9+10\end{bmatrix} |
= |
\begin{bmatrix}-6 & 26\\-1 & 19\end{bmatrix} |
(v) `BA=` |
\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix} |
\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix} |
= |
\begin{bmatrix}1×2+3×3 & 1×4+3×2\\ -2×2+5×3 & -2×4+5×2\end{bmatrix} |
= |
\begin{bmatrix}2+9 & 4+6\\-4+15 & -8+10\end{bmatrix} |
= |
\begin{bmatrix}11 & 10\\11 & 2\end{bmatrix} |
Question 2: Compute the following:
(i) |
\begin{bmatrix}a & b\\-b & a\end{bmatrix} |
+ |
\begin{bmatrix}a & b\\b & a\end{bmatrix} |
Solution:
= |
\begin{bmatrix}a+a & b+b\\-b+b & a+a\end{bmatrix} |
= |
\begin{bmatrix}2a & 2b\\0 & 2a\end{bmatrix} |
(ii) |
\begin{bmatrix}a^2+b^2 & b^2+c^2\\ a^2+c^2 & a^2+b^2\end{bmatrix} |
+ |
\begin{bmatrix}2ab & 2bc\\-2ac & -2ab\end{bmatrix} |
Solution:
= |
\begin{bmatrix}a^2+b^2+2ab & b^2+c^2+2bc\\a^2+c^2-2ac & a^2+b^2-2ab\end{bmatrix} |
= |
\begin{bmatrix}(a+b)^2 & (b+c)^2\\(a-c)^2 & (a-b)^2\end{bmatrix} |
(iii) |
\begin{bmatrix}-1 & 4 & -6\\8 & 5 & 16\\2 & 8 & 5\end{bmatrix} |
+ |
\begin{bmatrix}12 & 7 & 6\\8 & 0 & 5\\3 & 2 & 4\end{bmatrix} |
Solution:
= |
\begin{bmatrix}-1+12 & 4+7 & -6+6\\8+8 & 5+0 & 16+5 \\2+3 & 8+2 & 5+4\end{bmatrix} |
= |
\begin{bmatrix}11 & 11 & 0\\16 & 5 & 21\\5 & 10 & 9\end{bmatrix} |
(iv) |
\begin{bmatrix}cos^2x & sin^2x \\sin^2x & cos^2x\end{bmatrix} |
+ |
\begin{bmatrix}sin^2x & cos^2x\\cos^2x & sin^2x\end{bmatrix} |
Solution:
= |
\begin{bmatrix}cos^2x+sin^2x & sin^2x+cos^2x\\sin^2x+cos^2x & cos^2x+sin^2x\end{bmatrix} |
= |
\begin{bmatrix}1 & 1\\1 & 1\end{bmatrix} |