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Compute the characteristic equation of a $3 \times 3$ matrix That is incorrect The characteristic equation comes from computing the $\lambda$ values necessary to give you a matrix determinant equal to zero $$\begin {vmatrix} -\lambda -1 -1 \\ 1 2 - \lambda 1 \\ 1 1 2- \lambda \end {vmatrix} = 0 $$ Notice that for $\lambda = 1$ all three columns become identical and we can come up with two eigenvectors Coming up with the last eigenvalue
Order of operations for multiplying three matrices Matrix multiplication is associative, so you can do it in whichever order you like You can prove it by writing the matrix multiply in summation notation each way and seeing they match
Difference between dimension and rank of matrix This is late, and for others stumbling upon this post The dimension is related to rank However the rank is the number of pivots, and for a Homogenous system the dimension is the number of free variables There is a formula that ties rank, and dimension together If you think about what you can do with a free variable why it is a dimension will be understood This can get slightly more
How is a Generator Matrix for a (7, 4) Hamming code created? I think @Timmmm s answer below provides a clue to the real answer which I've not seen here The Wikipedia article does, though: All bit positions that are power of 2 are parity (P) bits (1,2,4,8, ), others are data (D) bits It also shows how to calculate the parity bits in those positions, which leads for a Hamming (7,4)-code to the equations as shown in the Wikipedia article
What is the dimension of a matrix? - Mathematics Stack Exchange The dimension of a vector space is the number of coordinates you need to describe a point in it Thus, a plane in $\mathbb {R}^3$, is of dimension $2$, since each point in the plane can be described by two parameters, even though the actual point will be of the form $ (x,y,z)$ If you take the rows of a matrix as the basis of a vector space, the dimension of that vector space will give you the