how to multiply matrices 3×3 & 2×2 with examples

We have already defined various types of matrices with some examples. In this article, we will show you how to multiply matrices 3×3 & 2×2 of the same order along with examples.

Two matrices can only be multiplied when the number of rows in the first matrix is equal to the number of columns in the second matrix. If the number of rows in the first matrix is not equal to the number of columns in the second matrix then the two matrices cannot be multiplied. You can also go through another topic “matrix multiplication properties” on our blog. Let us first take an example of (2×2) matrix:

\[Let\:one\:matrix\:be\:[A]=\begin{bmatrix}
a&b\\
c&d\\
\end{bmatrix}\]

\[and\:another\:matrix\:be\:[B]=\begin{bmatrix}
p&q\\
r&s\\
\end{bmatrix}\]

The matrix multiplication of the matrix [A] & matrix [B] will be done as follows;
  • The first element in row one of the new matrix formed will be = (a × p) + (b × r).
  • The second element in the first row formed of the new matrix formed will be = (a × q) + (b × s).
  • Similarly, the first element in the second row of the new matrix formed will be = (c × p) + (d × r).
  • And the second element in the row second of the new matrix formed will be = (c × q)+ (d × s).

Therefore the new matrix formed will be;

how to multiply matrices 3x3

how to multiply matrices 3×3 properly? Let the two (3×3)  matrices be;

\[[C]=\begin{bmatrix}
1&3&2\\
2&4&1\\
2&3&5\\
\end{bmatrix}\]

\[and\:[D]=\begin{bmatrix}
4&5&2\\
3&4&2\\
2&6&1\\
\end{bmatrix}\]

The matrix multiplication of matrix [C] and matrix [D] will be carried as follows:
  • First element in row one of the new matrix formed will be = (1 × 4) + (3 × 3) + (2 × 2) = 17.
  • Second element in the first row formed of the new matrix formed will be = (1 × 5) + (3 × 4) + (2 × 6) = 29.
  • The third element in row one of the new matrix formed will be = (1 × 2) + (3 × 2) + (2 × 1) = 10.
  • First element in the second row of the new matrix formed will be = (2 × 4) + (4 × 3) + (1 × 2) = 22.
  • Second element in the second row of the new matrix formed will be = (2 × 5) + (4 × 4) + (1 × 6). = 32.
  • Third element in the second row of the new matrix formed will be = (2 × 2) + (4 × 2) + (1 × 1) = 13.
  • First element in row third of the new matrix formed will be = (2 × 4) + (3 × 3) + (5 × 2) = 27.
  • Second element in row third of the new matrix formed will be = (2 × 5) + (3 × 4) + (5 × 6) = 52.
  • And the third element in row third of the new matrix formed will be = (2 × 2) + (3 × 2) + (5 × 1) = 15.
Thus the new matrix formed by multiplying matrix [C] & [D] will be as shown below:

\[[C]⋅[D]=\begin{bmatrix}
17&29&10\\
22&32&13\\
27&52&15\\
\end{bmatrix}\]

This is how to multiply matrices 3×3 and 2×2. You can also calculate matrix multiplication online. If you find any difficulty in understanding this article you can directly comment below or you can also contact us in the contact section.