Given matrices S and T below, which statement is true?
1 answer:
Two matrices are inverse of each other if and only if the product of two matrices is an Identity matrix.
So, in order to find if S and T are inverse of each other, we have to show that
ST = TS = I
where I is the identity matrix of order 2 x 2.
![ST= \left[\begin{array}{cc}4&11\\-3&-8\end{array}\right] \left[\begin{array}{cc}-8&-11\\3&4\end{array}\right] \\ \\ ST=\left[\begin{array}{cc}4(-8)+11(3)&4(-11)+11(4)\\-3(-8)+(-8)(3)&-3(-11)+(-8)(4)\end{array}\right] \\ \\ ST=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]](https://tex.z-dn.net/?f=ST%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%2611%5C%5C-3%26-8%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-8%26-11%5C%5C3%264%5Cend%7Barray%7D%5Cright%5D%20%20%5C%5C%20%20%5C%5C%20%0A%0AST%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%28-8%29%2B11%283%29%264%28-11%29%2B11%284%29%5C%5C-3%28-8%29%2B%28-8%29%283%29%26-3%28-11%29%2B%28-8%29%284%29%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%20%20%5C%5C%20%0A%0AST%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D)
Similarly TS also comes out to be:
![TS=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]](https://tex.z-dn.net/?f=TS%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D)
Since TS is equal to ST and both are equal to the Identity matrix, we can conclude that S and T are the inverse of each other and therefore, option B is the correct answer.
So, in order to find if S and T are inverse of each other, we have to show that
ST = TS = I
where I is the identity matrix of order 2 x 2.
Similarly TS also comes out to be:
Since TS is equal to ST and both are equal to the Identity matrix, we can conclude that S and T are the inverse of each other and therefore, option B is the correct answer.
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