# GATE1996-10

1k views
Let $A = \begin{bmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \text { and } B = \begin{bmatrix} b_{11} && b_{12} \\ b_{21} && b_{22} \end{bmatrix}$ be two matrices such that $AB=I$. Let $C = A \begin{bmatrix} 1 && 0 \\ 1 && 1 \end{bmatrix}$ and $CD =I$. Express the elements of $D$ in terms of the elements of $B$.

$AB = I$, B is equal to the inverse of $A$ and vice versa.

So, $B= A^{-1}$

Now $CD =I$, $C$ is equal to the inverse of $D$ and vice versa.

So, $D =C^{-1}$ ​​​​​​

​$=\left(A.\begin{bmatrix} 1& 0 \\ 1&1 \end{bmatrix}\right)^{-1}$

Remark: $(AB)^{-1} = B^{-1}A^{-1}$

​$=\begin{bmatrix} 1& 0 \\ 1&1 \end{bmatrix}^{-1}.A^{-1}$

​$=\begin{bmatrix} 1& 0 \\ {-1}&1 \end{bmatrix}.B$

​$=\begin{bmatrix} b_{11}& b_{12} \\ b_{21}-b_{11}&b_{22}-b_{12} \end{bmatrix}$

edited
0
which formula in last line?
1
^ Multiplication of matrices $I and B$  :)

AB=CD

let x=[ 1  0 ]

[ 1  1]

AB=AXD

D=inv(X) *B

ANS IS          [b11                b12  ]

[ b21-b11   b22-b12  ]

1 vote
Suppose D = $\begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{bmatrix}$

Now, CD = AB  where C= A$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$

so,  A$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ = AB  [here A will be cancel out because on bothside A is a matrix]

$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ . $\begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{bmatrix}$ = A$\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$

$\begin{bmatrix} d_{11} & d_{12} \\ d_{11} + d_{21} & d_{12} + d_{22} \end{bmatrix}$ = $\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$

two matrix equally same means they are element wise equal.by this we can easily calculate, D = $\begin{bmatrix} b_{11} & b_{12} \\ b_{21} - b_{11} & b_{22} - b_{12} \end{bmatrix}$
0
in 3rd line why have you not written D after substituting value of C ?
Let |1 0| is X.

|1 1|

C = AX

AB = I, A = 1/B

C = (1/B)*X

CD = (1/B)*XD = 1*I XD = B. // left multiplication by B

D = (1/X)*B. // left multiplication by 1/X

(1/X)*B = |1 -1| * |b11 b12|

|0 1|     |b21 b22|

= |(b11-b21) b12-b22|

| b21 b22 |

= D

edited
1

## Related questions

1
1.4k views
The matrices $\begin{bmatrix} \cos\theta && -\sin\theta \\ \sin\theta && \cos\theta \end{bmatrix}$ and $\begin{bmatrix} a && 0\\ 0&& b \end{bmatrix}$ commute under multiplication if $a=b \text{ or } \theta = n\pi, n$ an integer always never if $a \cos\theta = b \sin\theta$
Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknowns. Which of the following is false? The system has a solution if and only if, both $A$ ... system has a unique solution. The system will have only a trivial solution when $m=n$, $b$ is the zero vector and $\text{rank}(A) =n$.
Assume that the matrix $A$ given below, has factorization of the form $LU=PA$, where $L$ is lower-triangular with all diagonal elements equal to 1, $U$ is upper-triangular, and $P$ is a permutation matrix. For $A = \begin{bmatrix} 2 & 5 & 9 \\ 4 & 6 & 5 \\ 8 & 2 & 3 \end{bmatrix}$ Compute $L, U,$ and $P$ using Gaussian elimination with partial pivoting.
Let $F$ be the collection of all functions $f: \{1, 2, 3\} \to \{1, 2, 3\}$. If $f$ and $g \in F$, define an equivalence relation $\sim$ by $f\sim g$ if and only if $f(3) = g(3)$. Find the number of equivalence classes defined by $\sim$. Find the number of elements in each equivalence class.