Find Matrix A: Determining Inverse of Matrices

In summary, the question asks to determine matrix A, where A = ${S}^{-1}$ * B * S. If matrices B and S commute, then the answer would be A = B. However, in general, matrix multiplication is not commutative, so the best that can be done is to multiply on the left and right by S and ${S}^{-1}$, respectively, to get A = B.
  • #1
lyd123
13
0
Hi, I've a question that asks me to determine matrix A , where A= ${S}^{-1}$* B* S
They have given matrices S and B in the question.

I think the answer is A=B, since S * ${S}^{-1}$ would give me the identity matrix and anything multiplied by the identity matrix is itself. Is this correct?
 
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  • #2
Hi lyd123.

The answer $A=B$ would be correct only if the matrices $B$ and $S$ commute, i.e. only if $BS=SB$. Perhaps the matrices you are given commute. In general, however, matrix multiplication is not commutative. Given
$$A\ =\ S^{-1}BS$$
the best you can do is multiply on the left by $S$ and on the right by $S^{-1}$ to get
$$SAS^{-1}\ =\ S(S^{-1}BS)S^{-1}\ =\ (SS^{-1})B(SS^{-1})\ =\ IBI\ =\ B.$$
 
  • #3
Thank you! I totally forgot that matrix multiplication isn't commutative.
 

1. How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use the Gauss-Jordan elimination method, the adjugate method, or the Smith normal form method. These methods involve performing a series of row operations on the matrix to transform it into its inverse form.

2. What does it mean for a matrix to be invertible?

A matrix is invertible if it has an inverse, which means that there exists another matrix that, when multiplied with the original matrix, results in the identity matrix. In other words, the inverse of a matrix "undoes" the effects of the original matrix.

3. Can all matrices be inverted?

No, not all matrices can be inverted. A matrix must be square (equal number of rows and columns) and have a non-zero determinant in order to have an inverse. Matrices that do not meet these criteria are called singular or non-invertible matrices.

4. Why is it important to find the inverse of a matrix?

Finding the inverse of a matrix is important in many areas of mathematics and science. It allows us to solve systems of linear equations, calculate determinants, and perform transformations. Inverse matrices are also used in data compression, computer graphics, and cryptography.

5. Are there any shortcuts or tricks for finding the inverse of a matrix?

While there are no shortcuts or tricks for finding the inverse of a matrix, there are certain properties that can make the process easier. For example, if a matrix has a simple structure such as being diagonal or triangular, the inverse can be found more quickly. Additionally, there are computer algorithms that can efficiently find the inverse of large matrices.

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