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2.3: The ijth Entry of a Product

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In previous sections, we used the entries of a matrix to describe the action of matrix addition and scalar multiplication. We can also study matrix multiplication using the entries of matrices.

What is the ijth entry of AB? It is the entry in the ith row and the jth column of the product AB.

Now if A is m×n and B is n×p, then we know that the product AB has the form [a11a12a1na21a22a2nam1am2amn][b11b12b1jb1pb21b22b2jb2pbn1bn2bnjbnp]

The jth column of AB is of the form [a11a12a1na21a22a2nam1am2amn][b1jb2jbnj]

which is an m×1 column vector. It is calculated by b1j[a11a21am1]+b2j[a12a22am2]++bnj[a1na2namn]

Therefore, the ijth entry is the entry in row i of this vector. This is computed by ai1b1j+ai2b2j++ainbnj=nk=1aikbkj

The following is the formal definition for the ijth entry of a product of matrices.

Definition 2.3.1: The ijth Entry of a Product

Let A=[aij] be an m×n matrix and let B=[bij] be an n×p matrix. Then AB is an m×p matrix and the (i,j)-entry of AB is defined as (AB)ij=nk=1aikbkj

Another way to write this is (AB)ij=[ai1ai2ain][b1jb2jbnj]=ai1b1j+ai2b2j++ainbnj

In other words, to find the (i,j)-entry of the product AB, or (AB)ij, you multiply the ith row of A, on the left by the jth column of B. To express AB in terms of its entries, we write AB=[(AB)ij].

Consider the following example.

Example 2.3.1: The Entries of a Product

Compute AB if possible. If it is, find the (3,2)-entry of AB using Definition 2.3.1. A=[123126],B=[231762]

Solution

First check if the product is possible. It is of the form (3×2)(2×3) and since the inside numbers match, it is possible to do the multiplication. The result should be a 3×3 matrix. We can first compute AB: [[123126][27],[123126][36],[123126][12]]

where the commas separate the columns in the resulting product. Thus the above product equals [1615513155464214]
which is a 3×3 matrix as desired. Thus, the (3,2)-entry equals 42.

Now using Definition 2.3.1, we can find that the (3,2)-entry equals 2k=1a3kbk2=a31b12+a32b22=2×3+6×6=42

Consulting our result for AB above, this is correct!

You may wish to use this method to verify that the rest of the entries in AB are correct.

Here is another example.

Example 2.3.2: Finding the Entries of a Product

Determine if the product AB is defined. If it is, find the (2,1)-entry of the product. A=[231762000],B=[123126]

Solution

This product is of the form (3×3)(3×2). The middle numbers match so the matrices are conformable and it is possible to compute the product.

We want to find the (2,1)-entry of AB, that is, the entry in the second row and first column of the product. We will use Definition 2.3.1, which states (AB)ij=nk=1aikbkj

In this case, n=3, i=2 and j=1. Hence the (2,1)-entry is found by computing (AB)21=3k=1a2kbk1=[a21a22a23][b11b21b31]
Substituting in the appropriate values, this product becomes [a21a22a23][b11b21b31]=[762][132]=1×7+3×6+2×2=29

Hence, (AB)21=29.

You should take a moment to find a few other entries of AB. You can multiply the matrices to check that your answers are correct. The product AB is given by AB=[1313293200]


This page titled 2.3: The ijth Entry of a Product is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.

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