76.1 Matrix Addition and Subtraction Concept Overview

MATRIX ADDITION & SUBTRACTION | CONCEPT OVERVIEW The topic of MATRIX ADDITION & SUBTRACTION can be referenced on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

CONCEPT INTRO: A MATRIX is an ordered rectangular array of numbers made of a m rows and n columns. The NOMENCLATURE 𝑎!" , is an ELEMENT of the particular MATRIX, titled “a”, with the variables 𝑖 and 𝑗 referencing the ROW and the COLUMN, respectively, which that ELEMENT is located. The DIMENSION of a matrix is given in respect to the number of ROWS and COLUMNS, using the nomenclature 𝑚 𝑥 𝑛. This indicates that there are 𝑚 for the number of rows, and 𝑛 for the number of columns. There are many situations where we will want to use of MATRICES, one of which is to assess various SYSTEMS OF EQUATIONS. In this scenario, the ELEMENTS within the matrix would represent the coefficients of the set of equations being analyzed. The DIMENSION of the matrix will correspond to the number of equations represented in the set, which would define the number of ROWS, and the number of

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unique variables within the set, which would be represented by the number of COLUMNS. To ADD or SUBTRACT two matrices, the corresponding ELEMENTS from each of the MATRICES are COMBINED together to create a new unique matrix. A SQUARE MATRIX is one that has the SAME NUMBER of ROWS and COLUMNS, both of which are equal to the order of the matrix.

MATRIX ADDITION AND SUBTRACTION: The TOPIC of MATRIX ADDITION & SUBTRACTION can be referenced under the topic of MATRICES on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. ADDITION and SUBTRACTION of two matrices is POSSIBLY ONLY IF both MATRICES have equivalent DIMENSIONS, meaning that they are the same shape and size, ROWS and COLUMNS. The process of ADDING or SUBTRACTING MATRICES simply comes down to taking any of the ELEMENTS in one MATRIX and ADDING of SUBTRACTING that value with the corresponding ELEMENT in another MATRIX. The result is a new ELEMENT that is then placed in to the new MATRIX in the same location of the two elements used in this operation.



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The GENERAL FORMULA for the ADDITION of TWO MATRICES can be referenced under the topic of MATRICES on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The addition of matrices is represented by the general expression: 𝐴 𝐷

𝐵 𝐸

𝐺 𝐶 + 𝐽 𝐹

𝐻 𝐾

𝐼 𝐴+𝐺 = 𝐿 𝐷+𝐽

𝐵+𝐻 𝐸+𝐾

𝐶+𝐼 𝐹+𝐿

To illustrate this further, given Matrix “𝐴” and “𝐵” defined as: 𝑎!! 𝐴 = 𝑎!" 𝑎!"

𝑎!" 𝑎!! 𝑎!"

𝑎!" 𝑎!" 𝑎!"

𝑏!! 𝐵 = 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

𝑏!" 𝑏!" 𝑏!"

The addition of “𝐴” and “𝐵” would be: 𝑎!! 𝐴 + 𝐵 = 𝑎!" 𝑎!"

𝑎!" 𝑎!! 𝑎!"

𝑎!" 𝑏!! 𝑎!" + 𝑏!" 𝑎!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

𝑏!" 𝑎!! + 𝑏!! 𝑏!" = 𝑎!" + 𝑏!" 𝑏!" 𝑎!" + 𝑏!"

𝑎!" + 𝑏!" 𝑎!! + 𝑏!! 𝑎!" + 𝑏!"

𝑎!" + 𝑏!" 𝑎!" + 𝑏!" 𝑎!" + 𝑏!"

𝑏!" 𝑎!! − 𝑏!! 𝑏!" = 𝑎!" − 𝑏!" 𝑏!" 𝑎!" − 𝑏!"

𝑎!" − 𝑏!" 𝑎!! − 𝑏!! 𝑎!" − 𝑏!"

𝑎!" − 𝑏!" 𝑎!" − 𝑏!" 𝑎!" − 𝑏!"

The subtraction of “𝐴” and “𝐵” would be: 𝑎!! 𝐴 − 𝐵 = 𝑎!" 𝑎!"

𝑎!" 𝑎!! 𝑎!"

𝑎!" 𝑏!! 𝑎!" − 𝑏!" 𝑎!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

It is worth bringing up again that it is only possible to ADD or SUBTRACT two matrices that have the same DIMENSIONS, or in other words, are the same shape.

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This can be assumed since it is necessary that each element within the first matrix has a corresponding element in the second to complete the operation. As an example, say we are given Matrix “𝐴” and “𝐵” defined as:

𝑎!! 𝐴= 𝑎 !"

𝑎!" 𝑎!!

𝑎!" 𝑎!"

𝑏!! 𝐵 = 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

𝑏!" 𝑏!" 𝑏!"

The DIMENSION of MATRIX A is 2x3, while the DIMENSION of MATRIX B is 3x3. This is obvious through observation, but if asked to ADD or SUBTRACT these two MATRICES, it wouldn’t be possible. However, say we are given Matrix “𝐴” and “𝐵” defined as: 𝑎!! 𝐴= 𝑎 !"

𝑎!" 𝑎!!

𝑎!" 𝑎!"

𝐵=

𝑏!! 𝑏!"

𝑏!" 𝑏!!

𝑏!" 𝑏!"

The DIMENSION of MATRIX A is 2x3 and the DIMENSION of MATRIX B is 2x3, so we are able to move forward with ADDING these two MATRICES.



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MATRIX ADDITION & SUBTRACTION | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

The sum of the two matrices, as illustrated, is most close to: 0 9

1 8

7 12 6 B. 22 6 C. 12 1 D. 56 A.

2 6 + 7 3

5 4

4 5

6 6 9 1 9 6 12 4 6 6 12 12 1 12 12 −5

SOLUTION: The TOPIC of MATRIX ADDITION & SUBTRACTION can be referenced under the topic of MATRICES on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. ADDITION and SUBTRACTION of two matrices is POSSIBLY ONLY IF both MATRICES have equivalent DIMENSIONS, meaning that they are the same shape and size, ROWS and COLUMNS.

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The process of ADDING or SUBTRACTING MATRICES simply comes down to taking any of the ELEMENTS in one MATRIX and ADDING of SUBTRACTING that value with the corresponding ELEMENT in another MATRIX. The result is a new ELEMENT that is then placed in to the new MATRIX in the same location of the two elements used in this operation. The GENERAL FORMULA for the ADDITION of TWO MATRICES can be referenced under the topic of MATRICES on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The addition of matrices is represented by the general expression: 𝐴 𝐷

𝐵 𝐸

𝐺 𝐶 + 𝐽 𝐹

𝐻 𝐾

𝐼 𝐴+𝐺 = 𝐿 𝐷+𝐽

𝐵+𝐻 𝐸+𝐾

𝐶+𝐼 𝐹+𝐿

To illustrate this further, given Matrix “𝐴” and “𝐵” defined as: 𝑎!! 𝐴 = 𝑎!" 𝑎!"

𝑎!" 𝑎!! 𝑎!"

𝑎!" 𝑎!" 𝑎!"

𝑏!! 𝐵 = 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

𝑏!" 𝑏!" 𝑏!"

The addition of “𝐴” and “𝐵” would be: 𝑎!! 𝐴 + 𝐵 = 𝑎!" 𝑎!"



𝑎!" 𝑎!! 𝑎!"

𝑎!" 𝑏!! 𝑎!" + 𝑏!" 𝑎!" 𝑏!"

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𝑏!" 𝑏!! 𝑏!"

𝑏!" 𝑎!! + 𝑏!! 𝑏!" = 𝑎!" + 𝑏!" 𝑏!" 𝑎!" + 𝑏!"

𝑎!" + 𝑏!" 𝑎!! + 𝑏!! 𝑎!" + 𝑏!"

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𝑎!" + 𝑏!" 𝑎!" + 𝑏!" 𝑎!" + 𝑏!"

In this problem, we are given:

𝐴=

0 9

1 8

2 7

𝐵=

6 3

5 4

4 5

Through observation, we can confirm that the DIMENSION of MATRIX A is 2x3 and the DIMENSION of MATRIX B is 2x3. The SOLUTION in route to creating the resulting new MATRIX will run along these lines: 𝑎!! 𝐴+𝐵 = 𝑎 !"

𝑎!" 𝑎!!

𝑎!" 𝑏!! + 𝑎!" 𝑏!"

𝑏!" 𝑏!!

𝑏!" 𝑎 + 𝑏!! = !! 𝑏!" 𝑎!" + 𝑏!"

𝑎!" + 𝑏!" 𝑎!! + 𝑏!!

𝑎!" + 𝑏!" 𝑎!" + 𝑏!"

Now it’s just a matter of matching up the ELEMENTS and carrying out the ADDITION OPERATION at each location. Doing so we get:

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 5

Let’s walk through and HIGHLIGHT the first few ELEMENTS to illustrate specifically how MATRIX ADDITION is carried out. Starting with ELEMENT 𝑎!! and 𝑏!! , we will focus on the two values:

𝐴+𝐵 =

0 9

1 8

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5 4

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Which results in the OPERATION:

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 = 0+6 5

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 = 6 5

Or:

Now moving on to ELEMENTS 𝑎!" and 𝑏!" , we will focus on the two values:

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 = 6 5

Which results in the OPERATION:

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 = 6 5

1+5

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 = 6 5

6

Or:

Now moving on to ELEMENTS 𝑎!" and 𝑏!" , we will focus on the two values:

𝐴+𝐵 =



0 9

1 8

2 6 + 7 3 Made with

5 4

4 = 6 5

6

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Which results in the OPERATION:

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 6 = 5

6

2+4

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 6 = 5

6

6

Or:

Now filling out the remaining ELEMENT OPERATIONS, we get:

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

6 4 = 9+3 5

𝐴+𝐵 =

0 9

1 8

2 6 + 7 3

5 4

4 6 = 12 5

6 8+4

6 7+5

And: 6 12

6 12

This is our new MATRIX derived from the ADDITION of MATRIX A and MATRIX B.

The correct answer choice is C.



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