78.1 Matrix Multiplication Concept Overview

MATRIX MULTIPLICATION | CONCEPT OVERVIEW The topic of MATRIX MULTPLICATION 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. A very common MATRIX OPERATION that we will be required to know and perform come exam day, is that of MATRIX MULTIPLICATION. Given a pair of matrices, we are able to MULTIPLY the ELEMENTS from each of the matrices using a specific SYSTEMATIC approach, defining each ELEMENT in a new unique MATRIX in the process. Whereas in MATRIX ADDITION or SUBTRACTION, we need to be presented a pair of MATRICES having the same DIMENSIONS, or in other words, are the same shape, to proceed with the OPERATION. In MATRIX MULTIPLICATION, we need a set of MATRICES that share the NUMBER of COLUMNS in the leading MATRIX with the NUMBER of ROWS in the secondary MATRIX. To illustrate this further, if: Matrix A has the DIMENSIONS 𝑟 𝑥 𝑐 And: Matrix B has the DIMENSIONS 𝑐 𝑥 𝑚



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Then the NUMBER of COLUMNS in MATRIX A is the same as the NUMBER of ROWS in MATRIX B. Despite the other two DIMENSIONS not coordinating, MATRIX MULTIPLICATION can be carried out. Taking the DIMENSIONS defined in these two matrices, the result of MULTIPLYING MATRIX A and MATRIX B will be a new UNIQUE MATRIX with the DIMENSIONS 𝑟 𝑥 𝑚. If on the other hand: Matrix A has the DIMENSIONS 𝑟 𝑥 𝑐 And: Matrix B has the DIMENSIONS 𝑚 𝑥 𝑐 Then there is no COLUMN to ROW relationship established between the two MATRICES and MULTIPLICATON cannot be carried out.

MATRIX MULTIPLICATION: The GENERAL FORMULA for MULTIPLICATION of two MATRICES can be referenced under the topic of MATHEMATICS on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

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Given the MATRIX A and MATRIX B as:

𝑎!! 𝐴= 𝑎 !"

𝑎!" 𝑎!!

𝑎!" 𝑎!"

𝑏!! 𝐵 = 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

We can define MATRIX A as a 2x3 MATRIX and MATRIX B as a 3x2 MATRIX. In order for MULTIPLICATION to be possible on these two defined MATRICES, then we need to confirm a COLUMN to ROW relationship between them…meaning the NUMBER of COLUMNS in MATRIX A must EQUAL the NUMBER of ROWS in MATRIX B. If this is the case, which it is, we can calculate the product of MATRIX A and MATRIX B using a SYSTEMATIC approach illustrated as:

𝑎!! 𝐶= 𝑎 !"

𝑎!" 𝑎!! =

𝑏 𝑎!" !! 𝑎!" 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

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

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

It is important to note that MATRIX MULTIPLICATION is ANTI-COMMUTATIVE, as the ORDER of the MULTIPLICATION will produce an UNIQUE result.



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To illustrate this:

𝑎!! 𝑎!"

𝑎!" 𝑎!!

𝑏 𝑎!" !! 𝑎!" 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!! ≠ 𝑏!" 𝑏!" 𝑏!"

𝑏!" 𝑎 𝑏!! 𝑎!! !" 𝑏!"

𝑎!" 𝑎!!

𝑎!" 𝑎!"

ORDER does MATTER for the resulting MATRIX. It is worth noting again that in order for MULTIPLICATION to be possible then a COLUMN to ROW relationship between any two MATRICES must exist…meaning the NUMBER of COLUMNS in the leading MATRIX A must EQUAL the NUMBER of ROWS in the secondary MATRIX.



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MATRIX MULTIPLICATION | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. Carrying out the noted operation results in:

0 4

−1 11

3 2 1 2 6

−1 2 1

1 5 7 10 4 0 B. 76 20 15 20 C. 3 4 11 0 D. 35 20 A.

SOLUTION: The GENERAL FORMULA for MULTIPLICATION of two MATRICES can be referenced under the topic of MATHEMATICS on page 34 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Given a pair of matrices, we are able to MULTIPLY the ELEMENTS from each of the matrices using a specific SYSTEMATIC approach, defining each ELEMENT in a new unique MATRIX in the process.

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In MATRIX MULTIPLICATION, we need a set of MATRICES that share the NUMBER of COLUMNS in the leading MATRIX with the NUMBER of COLUMNS in the secondary MATRIX. Given the MATRIX A and MATRIX B as:

𝑎!! 𝐴= 𝑎 !"

𝑎!" 𝑎!!

𝑎!" 𝑎!"

𝑏!! 𝐵 = 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

We can define MATRIX A as a 2x3 MATRIX and MATRIX B as a 3x2 MATRIX. In order for MULTIPLICATION to be possible on these two defined MATRICES, then we need to confirm a COLUMN to ROW relationship between them…meaning the NUMBER of COLUMNS in MATRIX A must EQUAL the NUMBER of ROWS in MATRIX B. If this is the case, then we can move forward with calculating the product of MATRIX A and MATRIX B using a SYSTEMATIC approach illustrated as:

𝑎!! 𝐶= 𝑎 !"

𝑎!" 𝑎!! =



𝑏 𝑎!" !! 𝑎!" 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

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

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

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In this problem, we are given the OPERATION:

0 𝐴𝐵 = 𝐶 = 4

−1 11

3 2 1 2 6

−1 2 1

The first thing we must do is confirm that we can actually MULTIPLY these two MATRICES. By observation, we can see that we have a 2x3 MATRIX and a 3x2 MATRIX. The NUMBER of COLUMNS in MATRIX A is the same as the NUMBER of ROWSS in MATRIX B, therefore, MATRIX MULTIPLICATION can be carried out. Revisiting the SYSTEMATIC process of MATRIX MULTIPLICATION, we have:

𝑎!! 𝐶= 𝑎 !"

𝑎!" 𝑎!! =

𝑏 𝑎!" !! 𝑎!" 𝑏!" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

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

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

We will lay this process out step by step, defining a few elements from scratch, and then fill in the rest.



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Building on our template, we can define the ELEMENTS in MATRIX A as:

0 𝐶= 4

−1 11

𝑏!! 2 𝑏 2 !" 𝑏!"

𝑏!" 𝑏!! 𝑏!"

Doing the same for MATRIX B, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 2 1

This OPERATION will result in a new 2x2 MATRIX C, such that:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 𝑐!! 2 = 𝑐 !" 1

𝑐!" 𝑐!!

Let’s start off with defining ELEMENT 𝑐!! , in our resultant MATRIX C. We will proceed with defining this ELEMENT using the OPERATION:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 𝑎 𝑏 + 𝑎!" 𝑏!" + 𝑎!" 𝑏!" 2 = !! !! 𝑐!" 1

𝑐!" 𝑐!!

Let’s highlight each step in defining this specific ELEMENT starting with the term 𝑎!! 𝑏!! and ignoring everything else:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 𝑎 𝑏 + 𝑎!" 𝑏!" + 𝑎!" 𝑏!" 2 = !! !! 𝑐!" 1

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𝑐!" 𝑐!!

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We are being asked to MULTIPLY element 𝑎!! and 𝑏!! , which is:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 𝑎 𝑏 + 𝑎!" 𝑏!" + 𝑎!" 𝑏!" 2 = !! !! 𝑐!" 1

𝑐!" 𝑐!!

And filling out MATRIX C, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 0 3 + 𝑎!" 𝑏!" + 𝑎!" 𝑏!" 2 = 𝑐!" 1

𝑐!" 𝑐!!

Let’s cycle right to the next portion in defining this ELEMENT, which is:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 0 3 + 𝑎!" 𝑏!" + 𝑎!" 𝑏!" 2 = 𝑐!" 1

𝑐!" 𝑐!!

And filling in MATRIX C, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 0 3 + −1 1 + 𝑎!" 𝑏!" 2 = 𝑐!" 1

𝑐!" 𝑐!!

Cycling once more to the right:

0 𝐶= 4



−1 11

3 2 1 2 6

−1 0 3 + −1 1 + 𝑎!" 𝑏!" 2 = 𝑐!" 1

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𝑐!" 𝑐!!

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Filling in MATRIX C, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 0 3 + −1 1 + 2(6) 2 = 𝑐!" 1

𝑐!" 𝑐!!

Highlighting the full operation for defining ELEMENT 𝑐!! , we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 0 3 + −1 1 + 2(6) 2 = 𝑐!" 1

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐 !" 1

𝑐!" 𝑐!!

Or:

𝑐!" 𝑐!!

Let’s start off with defining ELEMENT 𝑐!" , in our resultant MATRIX C. We will proceed with defining this ELEMENT using the OPERATION:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐 !" 1

𝑎!! 𝑏!" + 𝑎!" 𝑏!! + 𝑎!" 𝑏!" 𝑐!!

Let’s highlight each step in defining this specific ELEMENT starting with the term 𝑎!! 𝑏!" and ignoring everything else:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐 !" 1

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𝑎!! 𝑏!" + 𝑎!" 𝑏!! + 𝑎!" 𝑏!" 𝑐!!

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We are being asked to MULTIPLY element 𝑎!! and 𝑏!" , which is:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

𝑎!! 𝑏!" + 𝑎!" 𝑏!! + 𝑎!" 𝑏!" 𝑐!!

And filling out MATRIX C, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

0(−1) + 𝑎!" 𝑏!! + 𝑎!" 𝑏!" 𝑐!!

Let’s cycle right to the next portion in defining this ELEMENT, which is:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

0 −1 + 𝑎!" 𝑏!! + 𝑎!" 𝑏!" 𝑐!!

And filling in MATRIX C, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

0 −1 + −1 (2) + 𝑎!" 𝑏!" 𝑐!!

Cycling once more to the right:

0 𝐶= 4



−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

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0 −1 + −1 (2) + 𝑎!" 𝑏!" 𝑐!!

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Filling in MATRIX C, we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

0 −1 + −1 (2) + 2(1) 𝑐!!

Highlighting the full operation for defining ELEMENT 𝑐!" , we have:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐!" 1

0 −1 + −1 (2) + 2(1) 𝑐!!

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 𝑐 !" 1

0 𝑐!!

Or:

So I think you see by now the TEMPLATE OPERATIONS in defining each ELEMENT within the resulting MATRIX C. This also highlights why it’s MANDATORY that the two matrices being MULTIPLIED share that COLUMN to ROW relationship. If they didn’t, the OPERATION would fail because ELEMENTS in one MATRIX will eventually not have any corresponding ELEMENT to MUTLIPLY against in a second MATRIX.



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Bypassing the combination of terms, filling in the remainder of the resultant MATRIX C gives us:

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 4 3 + 11 1 + 2(6) 1

0 𝐶= 4

−1 11

3 2 1 2 6

−1 11 2 = 35 1

0 4 −1 + 11 2 + 2(1)

Or:

The correct answer choice is D.



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0 20

𝟏𝟏 𝟑𝟓

𝟎 𝟐𝟎

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