Addition and Subtraction of Matrices
\nTo add or subtract matrices they must be of the same order. The components of the matrices can then be added or subtracted.
\nExample of addition of matrices- \\[\\begin {bmatrix}
1 & 7 & 9 \\\\
3 & 4 & 3
\\end {bmatrix} +
\\begin {bmatrix}
4 & 1 & 2 \\\\
5 & 6 & 6
\\end {bmatrix} =
\\begin {bmatrix}
1+4 & 7+1 & 9+2 \\\\
3+5 & 4+6 & 3+6
\\end{bmatrix} \\]
=
\n\\[\\begin {bmatrix}
5 & 8 & 11 \\\\
8 & 10 & 9
\\end{bmatrix}
\\]
Example of subtraction of matrices- \\[\\begin {bmatrix}
9 & 5 & 12 \\\\
10 & 4 & 3
\\end {bmatrix} -
\\begin {bmatrix}
3 & 2 & 6 \\\\
5 & 1 & 2
\\end {bmatrix} =
\\begin {bmatrix}
9-3 & 5-2 & 12-6 \\\\
10-5 & 4-1 & 3-2
\\end{bmatrix} \\]
=
\n\n
\\[\\begin {bmatrix}
6 & 3 & 6 \\\\
5 & 3 & 1
\\end{bmatrix}
\\]
Scalar multiplication of a matrix
\nTo multiply a matrix by a scalar simply multiply each individual element of the matrix by that number.
\nExample- \\[
2 \\times
\\begin {bmatrix}
4 & 2 &6 \\\\
5 & 7 & 3
\\end {bmatrix} =
\\begin {bmatrix}
8 & 4 & 12 \\\\
10 & 14 & 6
\\end {bmatrix}
\\]
Multiplying two matrices
\nThe number of columns in the first matrix has to be equal to the number of rows in the second matrix for multiplication to occur.
\nExample - \\[
\\begin {bmatrix}
2 & 4 & 2 \\\\
1 & 3 & 4
\\end {bmatrix} \\times
\\begin {bmatrix}
5 & 7 \\\\
3 & 6 \\\\
2 & 4
\\end {bmatrix} =
\\]
To start we need to multiply the first row and first column -
\n$ 2 \\times 5 +4 \\times 3 + 2 \\times 2 = 26 $
\nWe then multiply the first row by the second column -
\n$ 2 \\times 7 + 4 \\times 6 + 2 \\times 4 = 46$
\nWe then plug these two numbers into our answer matrix:
\n\\[\\begin {bmatrix}
26 & 46 \\\\
\\ddots & \\ddots
\\end{bmatrix}
\\]
We then multiply the second row and the first column -
\n$ 1 \\times 5 + 3 \\times 3 + 4 \\times 2 = 22$
\nWe then multiply the second row and the second column -
\n$ 1 \\times 7 + 3 \\times 6 + 4 \\times 4 = 41$
\nWe then insert these numbers into the answer matrix to complete the sum:
\n\\[\\begin {bmatrix}
26 & 46 \\\\
22 & 41
\\end{bmatrix}
\\]
Multiply these two matrices:
\n\\[
\\begin {bmatrix}
4 & 6 & 3 \\\\
2 & 9 & 7
\\end {bmatrix} \\times
\\begin {bmatrix}
5 & 3 \\\\
4 & 8 \\\\
2 & 5
\\end {bmatrix} =
\\]
Each section of the multiplication is broken down to help you.
", "advice": "After completing each multiplication step your finished matrix should look like this:
\n\\[
\\begin {bmatrix}
50 & 75 \\\\
60 & 113
\\end {bmatrix}
$ 4 \\times 5 + 6 \\times 4 + 3 \\times 2$ =
", "minValue": "50", "maxValue": "50", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$ 4 \\times 3 + 6 \\times 8 + 3 \\times 5$ =
", "minValue": "75", "maxValue": "75", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$ 2 \\times 5 + 9 \\times 4 + 7 \\times 2$ =
", "minValue": "60", "maxValue": "60", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$ 2 \\times 3 + 9 \\times 8 + 7 \\times 5$ =
", "minValue": "113", "maxValue": "113", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Complete the matrix with your multiplication answer
", "correctAnswer": "matrix([50,75],[60,113])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Becky Allen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3419/"}]}]}], "contributors": [{"name": "Becky Allen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3419/"}]}