Combining scalar multiples of matrices
A scalar combination is evaluated by scaling each matrix separately and then adding or subtracting the scaled matrices. The matrices must have the same order so that their matching positions can be combined
\[\left(sA\pm tB\right)_{ij}=sa_{ij}\pm tb_{ij}\]
- Check the order The two matrices must have the same number of rows and columns before their scalar multiples are combined \(\displaystyle A_{2\times2},\quad B_{2\times2}\quad\Rightarrow\quad 3A-2B\text{ is }2\times2\)
- Scale each matrix Multiply every entry of each matrix by the scalar attached to that matrix \(\displaystyle 3A-2B=3\begin{bmatrix}4&1\\2&3\end{bmatrix}-2\begin{bmatrix}1&2\\3&0\end{bmatrix}\)
- Match positions Combine entries with the same row and column number after scaling \(\displaystyle (3A-2B)_{12}=3a_{12}-2b_{12}\)
- Keep the signs A negative scalar changes the sign of every non-zero entry before the addition or subtraction is done. Zero entries stay zero \(\displaystyle (-2)(-1)-3(2)=2-6=-4\)
Worked example
Calculate 2A-3B for the two matrices shown below
\[A=\begin{bmatrix}3&-2\\1&4\end{bmatrix},\quad B=\begin{bmatrix}2&1\\-3&5\end{bmatrix}\]
- \(\displaystyle A\text{ and }B\text{ are both }2\times2\text{ matrices}\)The matrices have the same order, so the scalar combination is defined
- \(\displaystyle 2A=\begin{bmatrix}6&-4\\2&8\end{bmatrix},\quad 3B=\begin{bmatrix}6&3\\-9&15\end{bmatrix}\)Scale every entry by the scalar attached to its matrix
- \(\displaystyle 2A-3B=\begin{bmatrix}6-6&-4-3\\2-(-9)&8-15\end{bmatrix}\)Subtract corresponding entries of the two scaled matrices
- \(\displaystyle 2A-3B=\begin{bmatrix}0&-7\\11&-7\end{bmatrix}\)Evaluating each entry gives the final matrix
Common mistakes
Combining before scaling
✕\(\displaystyle (3A-2B)_{11}=3(4-1)=9\)✓\(\displaystyle (3A-2B)_{11}=3(4)-2(1)=10\)
Each matrix must be scaled by its own scalar before the two results are combinedAttaching scalars to the wrong matrices
✕\(\displaystyle \left(2A+3B\right)_{11}=3(1)+2(4)=11\)✓\(\displaystyle \left(2A+3B\right)_{11}=2(1)+3(4)=14\)
Each scalar is applied to the matrix immediately beside it before the two results are addedReversing a subtraction
✕\(\displaystyle 3(2)-2(5)=6-10=-4\)✓\(\displaystyle 2(5)-3(2)=10-6=4\)
For a difference, subtract the scaled second matrix from the scaled first matrix in the written orderDifficulty levels
introUse two positive scalars with a sum of two 2 × 2 matrices containing non-negative entries
standardUse a difference with a negative scalar, signed entries and a 3 × 2 or 3 × 3 order
challengeEvaluate two expressions on the same matrices, including one expression with an unscaled matrix and a zero answer entry
examChoose one entry of a scalar combination when the combine-first value is a distractor
Questions about this topic
Why are the matrices scaled before they are combined?+
The expression gives a separate scalar for each matrix, so each matrix must be scaled by its own scalar before corresponding entries are added or subtracted
When is a scalar combination of two matrices defined?+
The two matrices must have the same number of rows and the same number of columns
How is a negative scalar used?+
Multiply every entry by the negative scalar. Each non-zero entry changes sign, and each zero entry stays zero, before the matrices are combined
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals