Adding matrices entry by entry
Matrix addition is the friendliest operation of all: to add two matrices, just add the entries that sit in the same position. There is one rule — the two matrices must have the same dimensions, otherwise some entries would have no partner and the sum is undefined.
[ 2 -1 ] [ 4 5 ] [ 2+4 -1+5 ] [ 6 4 ] [ 0 3 ] + [ 1 -2 ] = [ 0+1 3-2 ] = [ 1 1 ] Same shape in, same shape out. Add matching positions.
Scalar multiplication: stretching a matrix
A scalar is just an ordinary number (to distinguish it from a matrix). Scalar multiplication means multiplying every entry of the matrix by that one number. Multiplying by 3 triples every entry; multiplying by −1 flips every sign, which is how we form the negative of a matrix and therefore how we subtract: A − B = A + (−1)B.
[ 2 -1 ] [ 3*2 3*(-1) ] [ 6 -3 ]
3 * [ 0 4 ] = [ 3*0 3*4 ] = [ 0 12 ]
Subtraction via scaling by -1:
[ 5 2 ] [ 1 3 ] [ 5 2 ] [ -1 -3 ] [ 4 -1 ]
[ 1 0 ] - [ 2 4 ] = [ 1 0 ] + [ -2 -4 ] = [-1 -4 ]The transpose: flip across the diagonal
The transpose of a matrix A, written A^T, turns its rows into columns and its columns into rows. The first row becomes the first column, the second row becomes the second column, and so on — like reflecting the grid across its main diagonal. A 2 × 3 matrix transposes into a 3 × 2 matrix; the dimensions swap.
[ 2 -1 0 ] [ 2 5 ]
A = [ 5 3 7 ] => A^T = [ -1 3 ]
(2 x 3) [ 0 7 ]
(3 x 2)
Row 1 of A -> Column 1 of A^T
The entry a(i,j) lands at position (j,i).