Matrices and Determinants · Mathematics · MHT CET

If $$A=\left[\begin{array}{ll}1 & -1 \\ 2 & -1\end{array}\right], B=\left[\begin{array}{cc}1 & 1 \\ 4 & -1\end{array}\right]$$, then $$(A+B)^{-1}$$ is...

Let $$A=\left[\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right].$$ If $$B=I-{ }^3 C_1(\operatorname{adj} A)+{ }^3 C_2(\operatorname{adj} A)^2-{ }^3 C...

If $$A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$$, then $$A^T \cdot A^{-1}=$$

If $$A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$ and $$A_{i j}$$ is a cofactor of $$a_{i j}$$ then the value of...

If $$A=\left[\begin{array}{cc}2 a & -3 b \\ 3 & 2\end{array}\right]$$ and $$A \cdot \operatorname{adj} A=A A^T$$, then $$2 a+3 b$$ is

If the matrix $$\mathrm{A}=\left[\begin{array}{cc}1 & 2 \\ -5 & 1\end{array}\right]$$ and $$\mathrm{A}^{-1}=x \mathrm{~A}+y \mathrm{I}$$, when $$I$$ i...

If $$\mathrm{A}=\left[\begin{array}{ll}\mathrm{i} & 1 \\ 1 & 0\end{array}\right]$$ where $$\mathrm{i}=\sqrt{-1}$$ and $$\mathrm{B}=\mathrm{A}^{2029}$$...

If $$P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$$ is the adjoint of a $$3 \times 3$$ matrix $$A$$ and $$|A|=...

Let $$\omega \neq 1$$ be a cube root of unity and $$S$$ be the set of all non-singular matrices of the form $$\left[\begin{array}{ccc}1 & a & b \\ \om...

If $$B=\left[\begin{array}{ccc}3 & \alpha & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3\end{array}\right]$$ is the adjoint of a $$3 \times 3$$ matrix $$\mathrm{A}$$...

If $$B=\left[\begin{array}{lll}1 & \alpha & 2 \\ 1 & 2 & 2 \\ 2 & 3 & 3\end{array}\right]$$ is the adjoint of a $$3 \times 3$$ matrix A and $$|A|=5$$,...

If $$\left|\begin{array}{ccc}\cos (A+B) & -\sin (A+B) & \cos (2 B) \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B\end{array}\right|=0$$, the...

Let $$A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1\end{array}\right], B=\left[\begin{array}{c}6 \\ 11 \\ 0\end{array}\right]$$ and $...

If $$A=\left[\begin{array}{cc}2 & -1 \\ -1 & 3\end{array}\right]$$, then the inverse of $$\left(2 A^2+5 A\right)$$ is

If $$A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 3\end{array}\right]$$ and $$B=\left[\begin{array}{ll}2 & 3 \\ 1 & 2 \\ 1 & 2\end{array}\right]$$, ...

Given $$A=\left[\begin{array}{ccc}x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z\end{array}\right]$$, if $$x y z=60$$ and $$8 x+4 y+3 z=20$$, then $$A$$.(adjA) ...

If $$\mathrm{A}=\left[\begin{array}{cc}\lambda & \mathrm{i} \\ \mathrm{i} & -\lambda\end{array}\right]$$ and $$\mathrm{A}^{-1}$$ does not exist, then ...

If $$A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$, and $$A(\operatorname{adj} A)=k I$$, then the value of $$(k+1...

IF $$A X=B$$, where $$A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array...

$$\text { If } A=\left[\begin{array}{ll} 2 & -2 \\ 2 & -3 \end{array}\right], B=\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right] \text {, th...

If $$A=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 1 & a \\ 2 & 4 & 7\end{array}\right]$$ and $$B=\left[\begin{array}{ccc}13 & 2 & b \\ -3 & -1 & 2 \\ -2...

For a $$3 \times 3$$ matrix $$\mathrm{A}$$, if $$\mathrm{A}(\operatorname{adj} \mathrm{A})=\left[\begin{array}{ccc}-10 & 0 & 0 \\ 0 & -10 & 2 \\ 0 & 0...

If $$A=\left[\begin{array}{ccc}5 & 6 & 3 \\ -4 & 3 & 2 \\ -4 & -7 & 3\end{array}\right]$$, then cofactors of all elements of second row are respective...

Which of the following matrices are invertible? $$\begin{aligned} & \mathrm{A}=\left[\begin{array}{cc} 2 & 3 \\ 10 & 15 \end{array}\right], \mathrm{B}...

If $$A=\left[\begin{array}{rr}2 & 3 \\ 5 & -2\end{array}\right]$$ and $$A^{-1}=K A$$, then $$K$$ is

If $$\mathrm{A}=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$, then $$\mathrm{A}(\operatorname{adj} \mathrm{A})=$$...

If $$A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right], B=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$$ and $...

If $$A=\left[\begin{array}{cc}5 a & -b \\ 3 & 2\end{array}\right]$$ and $$A$$ adj $$A=A A^T$$, then $$5 a+b=$$

For an invertible matrix $$A$$, if $$A(\operatorname{adj} A)=\left[\begin{array}{cc}20 & 0 \\ 0 & 20\end{array}\right]$$, then $$|A|=$$

If $$A=\left[\begin{array}{ccc}1 & 2 & 1 \\ -1 & 1 & 3\end{array}\right]$$ and $$B=\left[\begin{array}{cc}1 & 2 \\ -3 & 1 \\ 0 & 2\end{array}\right]$$...

If $$A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1\end{array}\right]$$, then the value of determinant of $$A^{-1}$$ is

If $$A = \left[ {\matrix{ k & 2 \cr { - 2} & { - k} \cr } } \right]$$, then A$$^{-1}$$ does not exists if k =

The sum of three numbers is 6. Thrice the third number when added to the first number gives 7. On adding three times first number to the sum of second...

If $$A=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$$, then $$\operatorname{...

If $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1\end{array}\right]$$ and $$A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -...

If $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$$, then $$A^{-1}=$$

If $$F(\propto)=\left[\begin{array}{ccc}\cos \propto & -\sin \propto & 0 \\ \sin \propto & \cos \propto & 0 \\ 0 & 0 & 1\end{array}\right]$$, where $$...

If $$A=\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right], \operatorname{adj} A=\left[\begin{array}{ccc}5 & x & -2 \\ 1 &...

$$\mathrm{A}^{-1}=\frac{-1}{2}\left[\begin{array}{cc}1 & -4 \\ -1 & 2\end{array}\right]$$, then $$2 A+I_2=\quad$$ where $$I_2$$ is a unit matrix of o...

The co-factors of the elements of second column of $$\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]$$ are

If $$A^{-1}=\left[\begin{array}{lll}3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 5 & 5\end{array}\right]$$, then $$A=$$

If $$A^{-1}=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$$ and $$B^{-1}=\left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]$$, then ...

$$A(\propto)=\left[\begin{array}{cc}\cos \propto & \sin \propto \\ -\sin \propto & \cos \propto\end{array}\right]$$, then $$\left[A^2(\propto)\right]^...

If inverse of $$\left[\begin{array}{ccc}1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6\end{array}\right]$$ does not exist, then $$x=$$

If $$A = \left[ {\matrix{ 3 & 2 & 4 \cr 1 & 2 & 1 \cr 3 & 2 & 6 \cr } } \right]$$ and A$$_{ij}$$ are cofactors of the elements a$$_{i...

If $A=\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]$ and $A^2-5 A-6 I=0$, then $A^{-1}=$

The cofactors of the elements of the first column of the matrix $A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 3 & 1 & 2 \\ -1 & 1 & 2\end{array}\right]$ ar...

The matrix $$A=\left[\begin{array}{rrr}a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2\end{array}\right]$$ is not invertible only if $$a=$$

If $$A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], \quad B=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]$$, then $$B^{-1} A^{-1...

The sum of the cofactors of the elements of second row of the matrix $$\left[\begin{array}{rrr}1 & 3 & 2 \\ -2 & 0 & 1 \\ 5 & 2 & 1\end{array}\right]$...

If $$A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$$ and $$A^{-1}=\left[\begin{array}{rrr}3 & -1 & 1 \\ \alpha & 6 ...

If $A$ and $B$ are square matrices of order 3 such that $|A|=2,|B|=4$, then $|A(\operatorname{adj} B)|=\ldots$

If $A$ is non-singular matrix and $(A+I)(A-I)=0$ then $A+A^{-1}=$ .............

If $A=\left[\begin{array}{cc}1+2 i & i \\ -i & 1-2 i\end{array}\right]$, where $i=\sqrt{-1}$, then $A(\operatorname{adj} A)=\ldots$

If $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$ and $A=A^{-1}$, then $x=\ldots \ldots$

If $A$ is non-singular matrix such that $(A-2 l)(A-4 I)=0$ then $A+8 A^{-1}=$ ..........