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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
Why does a determinant of $0$ mean the matrix isnt invertible? I always got taught that if the determinant of a matrix is $0$ then the matrix isn't invertible, but why is that? My flawed attempt at understanding things: This approaches the subject from a
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
What exactly is a matrix? - Mathematics Stack Exchange That is actually a lot of questions that might need thorough research for complete answers but as Lord Shark noticed, many things are explained by the linear algebra point of view (especially the multiplication) If you see a matrix as a set of vectors (its columns) then the determinant is the volume that is "defined" by those vectors They are used to solve linear algebra problems (among many
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