Show that if a matrix A is invertible, then A is non-singular. - Vidyadip

Question

Show that if a matrix A is invertible, then A is non-singular.

✓

Answer

Writing the above statement in symbolic form, we have p ⇒ q, where, p is “matrix A is invertible” and q is “A is non singular”
Instead of proving the given statement, we prove its contrapositive statement, i.e., if A is not a non singular matrix, then the matrix A is not invertible.
If A is not a non singular matrix, then it means the matrix A is singular, i.e.,
|A| = 0
Then $\mathrm{A}^{-1}=\frac{a d j \mathrm{A}}{| \mathrm{Al}} \text { does not exist as } \mathrm{|A|}=0$
Hence, A is not invertible.
Thus, we have proved that if A is not a non singular matrix, then A is not invertible.
i.e -q = -p
Hence, if a matrix A is invertible, then A is non singular.

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 "5 Marks Questions" from Appendix 2→Appendix 2 — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\text{y}=\cos^{-1}\Big\{\frac{2\text{x}-3\sqrt{1-\text{x}^2}}{\sqrt{13}}\Big\},$ find $\frac{\text{dy}}{\text{dx}}.$

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}|\text{x}-\text{a|}\sin(\frac{1}{\text{x}-\text{a}}), &\text{for} \text{ x} \neq\text{a}\\0,&\text{for} \text{ x} = \text{a}\end{cases}\text{ at x}=0$

Show that the relation R defined by R = {(a, b): a - b is divisible by 3; a, b ∈ Z} is an equivalence relation.

Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of kings.

Evaluate the following integrals:
$\int(\text{e}^{\log\text{x}}+\sin\text{x})\cos\text{x dx}$

Evaluate the following integrals:
$\int\frac{1}{(\text{x}^2-1)\sqrt{\text{x}^2+1}}\text{ dx}$

Find the equation of the plane through the points $(3, 4, 1)$ and $(0, 1, 0)$ and parallel to the line $\frac{\text{x}+3}{2}=\frac{\text{y}-3}{7}=\frac{\text{z}-2}{5}.$

By using properties of determinants, show that: $\begin{vmatrix}a^2+1&ab&ac\\ab&b^2+1&bc\\ca&cb&c^2+1\end{vmatrix}=1+a^2+b^2+c^2$

A and B has two families. Family A has 4 male, 6 female and 2 children and family B has 2 male, 2 female and 4 children. Every male, female and children has suggested to give respectively 2400, 1900 and 1800 calories per day and 45,55 and 33 gm. protein also. From this informa tion prepare a matrix. Using multiplication of matrix, find the total quantity of calorie and protein for every family.

Evaluate the following integrals:
$\int\limits^1_{-1}|\text{x}\cos\pi\text{x}|\text{dx}$