Question

01. Two matrixes A and B are given. Express B^-1 through x and A.

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Answer 1

Question :-

Two matrices A and B are given. Express B⁻¹ through 'x' and A.

$A=\begin{bmatrix}3x & -4x & 2x\\ -2x & x &0 \\ -x& -x & x\end{bmatrix}$

$B=\begin{bmatrix}x & 2x & -2x\\ 2x & 5x & -4x \\ 3x& 7x & -5x\end{bmatrix}$

Solution :-

$A=\begin{bmatrix}3x & -4x & 2x\\ -2x & x &0 \\ -x& -x & x\end{bmatrix}$

$A=x\begin{bmatrix}3& -4 & 2\\ -2 & 1 &0 \\ -1& -1 & 1\end{bmatrix}$

$B=\begin{bmatrix}x & 2x & -2x\\ 2x & 5x & -4x \\ 3x& 7x & -5x\end{bmatrix}$

$B=x\begin{bmatrix}1 & 2 & -2\\ 2 & 5 & -4 \\ 3& 7 & -5\end{bmatrix}$

Multiplication of Matrices of order 3 × 3,

$\begin{bmatrix}a & b & c\\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}j & k & l\\ m & n & o \\ p & q & r\end{bmatrix}=$

$\begin{bmatrix}aj+bm+cp & ak+bn+cq & al+bo+cr\\ dj+em+fp & dk+en+fq & dl+eo+fr \\ gj+hm+ip & gk+hn+iq & gl+ho+ir\end{bmatrix}$

Multiplying matrices A and B,

$AB=x\begin{bmatrix}3& -4 & 2\\ -2 & 1 &0 \\ -1& -1 & 1\end{bmatrix}x\begin{bmatrix}1 & 2 & -2\\ 2 & 5 & -4 \\ 3& 7 & -5\end{bmatrix}$

$AB=x^2\begin{bmatrix}3& -4 & 2\\ -2 & 1 &0 \\ -1& -1 & 1\end{bmatrix}\begin{bmatrix}1 & 2 & -2\\ 2 & 5 & -4 \\ 3& 7 & -5\end{bmatrix}$

$AB=x^2\begin{bmatrix}3(1)-4(2)+2(3) & 3(2)-4(5)+2(7) & 3(-2)-4(-4)+2(-5)\\ -2(1)+1(2)+0(3) & -2(2)+1(5)+0(7) & -2(-2)+1(-4)+0(-5) \\ -1(1)-1(2)+1(3) & -1(2)-1(5)+1(7) & -1(-2)-1(-4)+1(-5)\end{bmatrix}$

$AB=x^2\begin{bmatrix}3-8+6 & 6-20+14 & -6+16-10\\ -2+2+0 & -4+5+0 & 4-4+0 \\ -1-2+3 & -2-5+7 & 2+4-5\end{bmatrix}$

$AB=x^2\begin{bmatrix}1 & 0 & 0\\ 0 & 1 &0 \\ 0 & 0 & 1\end{bmatrix}$

$AB=x^2I,where\:I\:is\:Identity\:Matrix\:of\:order\:3\times3.$

$A=x^2B^{-1}$

$B^{-1}=x^{-2}A$

Answer :-

$\underline{\boxed{B^{-1}=x^{-2}A}}$

Extra Points to Remember :-

The necessary and sufficient condition for a square matrix A to be invertible  is |A| ≠ 0.

If A and B are invertible matrices of same order, then (AB)⁻¹ = B⁻¹A⁻¹.

If A is invertible then, (A⁻¹)⁻¹ = A.

If A is invertible then, (Aⁿ)⁻¹ = A⁻ⁿ, where n ∈ N.

Answer 2

Answer:

See the attachment ...

Vaiya it's the answer of your last question [BUET :18 -19].

Step-by-step explanation:

Hope it helps you :)

P.S :There was no space to answer in that question so I answered here....