Combine scalar multiples of matrices

Addition and subtraction require equal dimensions

For m×n matrices A and B, (A+B)ᵢⱼ=aᵢⱼ+bᵢⱼ and (A−B)ᵢⱼ=aᵢⱼ−bᵢⱼ. A 2×3 and a 3×2 matrix cannot be added. Matching physical labels and units are needed in a data application. The zero matrix of the same dimensions is the additive identity

Scale all entries, including negative ones

A scalar k multiplies every matrix entry. For the matrices A and B, first scale each matrix separately. Then combine corresponding entries

Given matrix

\(\displaystyle \begin{gathered}A=\begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}\\B=\begin{bmatrix} 1 & 4 \\ -2 & 5 \end{bmatrix}\end{gathered}\)

For these A and B, find the entry in row 1, column 2 of 2A−3B

Given matrix

\(\displaystyle \begin{gathered}A=\begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}\\B=\begin{bmatrix} 1 & 4 \\ -2 & 5 \end{bmatrix}\end{gathered}\)

For the matrices A and B, find the entry in row 2, column 1 of A+B.

Which expression equals 2A−3B for every same-order A and B?

Entrywise laws come from scalar arithmetic

Addition is commutative and associative. Scalar multiplication distributes over matrix sums, and (s+t)A=sA+tA. Subtraction is generally not commutative. These statements hold for equal-order matrices over the real numbers

Prove a matrix law, ungraded

Prove k(A+B)=kA+kB using an arbitrary entry. State the dimensions and explain why checking all entries proves matrix equality. Independent proof is ungraded

Model: prove the statement at every position

Let A and B both have order m×n and k be a real scalar. At arbitrary indices i=1..m,j=1..n, the left entry is k(aᵢⱼ+bᵢⱼ), which scalar distributivity makes kaᵢⱼ+kbᵢⱼ, the right entry. Both matrices have order m×n and every corresponding entry agrees, so the matrices are equal