Matrix addition is really easy—don't overthink it! All you need to do is combine the two matrices in a one-to-one manner for each index:

\(\displaystyle \begin{bmatrix} -5&7\\ 102 & -12 \end{bmatrix}+\begin{bmatrix} 3&107\\ -10 & -9 \end{bmatrix}=\begin{bmatrix} -5+3&7+107\\ 102-10 & -12-9 \end{bmatrix}\)

Then, just simplify all of those simple additions and subtractions:

\(\displaystyle \begin{bmatrix} -5+3&7+107\\ 102-10 & -12-9 \end{bmatrix}=\begin{bmatrix} -2&114\\ 92 & -21 \end{bmatrix}\)

To find the sum of two matrices, we simply add each entry from one matrix to the corresponding entry of the other matrix, and the result becomes the entry in the same location of the matrix for their sum:

\(\displaystyle \begin{bmatrix} 7& 16& -3\\ -13& 5&8 \\ 4&-2 &21 \end{bmatrix}+\begin{bmatrix} -11& 13& 4\\ 6& 5& -18\\ 16& 3& -9 \end{bmatrix}=\begin{bmatrix} -4& 29& 1\\ -7& 10& -10\\ 20& 1& 12 \end{bmatrix}\)

Since the dimensions of the two matrices are equal the sum of the two matrices exists.

To find the sum, add each component entry from the first matrix to the same component entry of the second matrix.

\(\displaystyle A+B=\begin{pmatrix} 5&1 \\ 3&-7 \end{pmatrix}+\begin{pmatrix} 4&2 \\ 9& 0 \end{pmatrix}}=\begin{pmatrix} 9&3 \\ 12&-7 \end{pmatrix}\)

Because the dimensions of the two matrices are not equal

(A 2x2 matrix is not of the same dimension as a 3x2 matrix)

The sum does not exist.

To add matricies, first check to make sure they are the same size. If they are, then just add corresponding terms in the same spots in each matrix.

Since these are the same size we can add as follows.

\(\displaystyle \begin{bmatrix} a &b &c \\d & e& f\\g &h &i \end{bmatrix}+\begin{bmatrix} j & k &l \\ m& n& o\\p &q &r \end{bmatrix}=\begin{bmatrix} a+j &b+k &c+l \\d+m & e+n& f+o\\g+p &h+q &i+r \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 6 &-4 &1 \\2 & 6& 5\\-1 &-3 &8 \end{bmatrix}+\begin{bmatrix} 1 & 4 &8 \\ 6& 3& -2\\3 &-2 &7 \end{bmatrix}=\begin{bmatrix} 6+1&-4+4 &1+8 \\2+6& 6+3& 5+-2\\-1+3 &-3+-2 &8+7 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 7 &0 &9 \\8 &9 &3 \\2 &-5 &15 \end{bmatrix}\)

Because the two matrices have the same dimension, the sum exists

\(\displaystyle \small \small \begin{pmatrix} 1&3 \\ 3&2 \end{pmatrix} +\begin{pmatrix} 0&1 \\ 1& 1 \end{pmatrix}= \begin{pmatrix} 1+0&3+1 \\ 3+1&2+1 \end{pmatrix}= \begin{pmatrix} 1&4 \\ 4&3 \end{pmatrix}\)

To add matricies, simply add the corresponding terms. Thus,

\(\displaystyle \begin{bmatrix} 3 &-1 \\-2 &4 \end{bmatrix} + \begin{bmatrix} -2 &0 \\2 &3 \end{bmatrix}=\begin{bmatrix} 3+(-2)& -1+0\\-2+2 &4+3 \end{bmatrix}=\begin{bmatrix} 1 &-1 \\0 &7 \end{bmatrix}\)

To add two matrices, combine the corresponding numbers in the matrices: (\(\displaystyle 1+0=1, 3+5=8, 2+(-2)=0, -1+6=5\)). Then, put those answers in the corresponding spots in the answer matrix. Therefore, your answer is \(\displaystyle \begin{bmatrix} 1 &8 \\ 0&5 \end{bmatrix}\).

Because the matrices do not have the same dimension, the sum does not exist.

Given a scalar k and a vector v, the vector given by their products is defined component-wise:

 \(\displaystyle k< v_1,v_2,...,v_n>=< kv_1,kv_2,...,kv_n>\).

Here, our product is:

\(\displaystyle \frac{4}{7}< -3,5,14>=< \frac{4}{7}(-3),\frac{4}{7}(5),\frac{4}{7}(14)>\\ \\ =< \frac{-12}{7},\frac{20}{7},8>\)