Download App

Matrices (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Matrices (p-1)5 Marks
Question
If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0\end{array}\right]$ thenshow that $A B$ and $BA$ are both singular matrices.
✓

Answer

$
\begin{aligned}
& A B=\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & 4 & 6 \\
1 & 2 & 3
\end{array}\right]\left[\begin{array}{rrr}
1 & -1 & 1 \\
-3 & 2 & -1 \\
-2 & 1 & 0
\end{array}\right] \\
&=\left[\begin{array}{rrr}
1-6-6 & -1+4+3 & 1-2+0 \\
2-12-12 & -2+8+6 & 2-4+0 \\
1-6-6 & -1+4+3 & 1-2+0
\end{array}\right] \\
&=\left[\begin{array}{rrr}
-11 & 6 & -1 \\
-22 & 12 & -2 \\
-11 & 6 & -1
\end{array}\right] \\
& \therefore| AB |=\left[\begin{array}{rrr}
-11 & 6 & -1 \\
-22 & 12 & -2 \\
-11 & 6 & -1
\end{array}\right]
\end{aligned}
$
By taking -6 common from $C_2$, we get
$
\begin{aligned}
& \therefore| AB |=-6\left|\begin{array}{lll}
-11 & -1 & -1 \\
-22 & -2 & -2 \\
-11 & -1 & -1
\end{array}\right| \\
& =-6 \times 0 \\
& \text {... }\left[\because C_2 \equiv C_3\right] \\
& =0 \\
&
\end{aligned}
$
$\therefore AB$ is a singular matrix.
Also, $BA =\left(\begin{array}{rrr}1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0\end{array}\right)\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3\end{array}\right)$
$
=\left(\begin{array}{rrr}
1-2+1 & 2-4+2 & 3-6+3 \\
-3+4-1 & -6+8-2 & -9+12-3 \\
-2+2+0 & -4+4+0 & -6+6+0
\end{array}\right)
$
$
=\left(\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)=0 \quad \therefore| BA |=0
$
$\therefore BA$ is also a singular matrix.
Hence, $AB$ and $BA$ are both singular matrices.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Matrices (p-1)→Matrices (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Solve the following equations by the method of inversion : $x + y + z = 1, x – y + z = 2$ and $x + y – z = 3$
→
Find the area between the parabolas $y^2=7 x$ and $x^2=7 y$.
→
Assuming the following statements
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth values of the following:
(i) Stock prices are not high or stocks are rising.
(ii) Stock prices are high and stocks are rising if and only if stock prices are high.
(iii) If stock prices are high, then stocks are not rising.
(iv) It is false that stocks are rising and stock prices are high.
(v) Stock prices are high or stocks are not rising iff stocks are rising.
→
Maximize z = 6x + 10y, subject to 3x + 5y ≤ 10, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0
→
Which of the following sentences are statements? In case of a statement, write down the truth value:
(i) The Himalayas is the highest mountain range.
(ii) $(x – 2)(x – 3) = x^2 – 5x + 6$ for all $x \in R.$
(iii) What are the causes of rural unemployment?
(iv) $0! = 1.$
(v) The quadratic equation $ax^2 + bx + c = 0 (a \neq 0)$ always has two real roots.
→
The rate of growth of the population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1 lakh, when will the city have a population of 400000 ?
→
Evalute : $\int \frac{3 e^x+4}{2 e^x-8} d x$
→
The data obtained on X, the length of time in weeks that a promotional project has been in progress at a small business, and Y the percentage increase in weekly sales over the period just prior to the beginning of the campaign.
X123413123424
Y101018201115121517191316
Find the equation of regression line to predict percentage increase in sales if the company has been in progress for 1.5 weeks.
→
There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subjects - Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices $A$ and $B$ :
July sales (in Rupees), Physics, Chemistry, Mathematics $A=\left[\begin{array}{lll}5600 & 6750 & 8500 \\ 6650 & 7055 & 8905\end{array}\right]$ First Row Suresh / Second Row Ganesh
August Sales (in Rupees), Physics, Chemistry, Mathematics $B =\left[\begin{array}{ccc}6650 & 7055 & 8905 \\ 7000 & 7500 & 10200\end{array}\right]$ First Row Suresh / Second Row Ganesh
(i) Find the increase in sales in Z from July to August 2017.
(ii) If both book shops get $10 \%$ profit in the month of August 2017, find the profit for each bookseller in each subject in that month.
→
Find the equations of tangent and normal to the following curves : $y=x^3-x^2-1$ at the point whose abscissa is -2 .
→