Matrix Operations

Scalar Multiplication

• The product of a real number and a matrix.
• Each entry in the matrix is multiplied by the given scalar.

Example:

$$n\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]=\left[\begin{array}{ccc}na& nb& nc\\ nd& ne& nf\end{array}\right]$$

Example 2:

$$2\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]=\left[\begin{array}{cc}2\cdot 1& 2\cdot 3\\ 2\cdot 4& 2\cdot 2\end{array}\right]=\left[\begin{array}{cc}2& 6\\ 8& 4\end{array}\right]$$

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Matrix Addition

• Add corresponding entries together.

Example:

$$\left[\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right]+\left[\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right]=\left[\begin{array}{cc}a+g& b+h\\ c+i& d+j\\ e+k& f+l\end{array}\right]$$

Example 2:

$$\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]+\left[\begin{array}{cc}5& 7\\ 6& 8\end{array}\right]=\left[\begin{array}{cc}1+5& 3+7\\ 4+6& 2+8\end{array}\right]=\left[\begin{array}{cc}6& 10\\ 10& 10\end{array}\right]$$

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Matrix Multiplication

• For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The result matrix has the number of rows of the first and the number of columns of the second matrix.

Example:

$$\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]\times \left[\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right]=\left[\begin{array}{cc}ag+bi+ck& ah+bj+cl\\ dg+ei+fk& dh+ej+fl\end{array}\right]$$

Example 2:

$$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\times \left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}1\cdot 5+2\cdot 7& 1\cdot 6+2\cdot 8\\ 3\cdot 5+4\cdot 7& 3\cdot 6+4\cdot 8\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$$

Scalar Multiplication (1x1 Matrices)

Multiplying 1x1 matrices is the same as multiplying scalars.

• Example:
  • Let ( A = [a] ) and ( B = [b] ) be 1x1 matrices (scalars).
  • The product, ( C = A \times B ), is ( C = [ab] ).
  • For instance, if ( A = [3] ) and ( B = [2] ), then ( C = [3 \times 2] = [6] ).

Multiplication of 2x2 Matrices

To multiply 2x2 matrices, we sum the products of corresponding elements from the rows of the first matrix and the columns of the second matrix.

• Example:
  • Let ( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} ) and ( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} ).
  • The product ( C = A \times B ) is calculated as: a_{11} \cdot b_{11} + a_{12} \cdot b_{21} & a_{11} \cdot b_{12} + a_{12} \cdot b_{22} \\ a_{21} \cdot b_{11} + a_{22} \cdot b_{21} & a_{21} \cdot b_{12} + a_{22} \cdot b_{22} \end{bmatrix} $$
  • For example, with $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$ the product is:
  1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$

Multiplication of 3x3 Matrices

The principle is the same as for 2x2 matrices, but involves three elements in each sum.

• Example:
  • Let ( A ) and ( B ) be 3x3 matrices.
  • The product ( C = A \times B ) is determined by multiplying and summing corresponding row and column elements.
  • This can be visualized in a similar manner to the 2x2 matrix multiplication.

Multiplication of Mismatched Matrices (2x3 and 3x2)

To multiply matrices of different sizes, the number of columns in the first matrix must equal the number of rows in the second. The resultant matrix has the number of rows of the first matrix and the number of columns of the second matrix.

• Example:
  • Let ( A ) be a 2x3 matrix and ( B ) be a 3x2 matrix.
  • The product ( C = A \times B ) will be a 2x2 matrix.
  • Each element of ( C ) is computed by multiplying corresponding elements and summing them up.

Determining the Size of the Output Matrix

• The size of the resulting matrix is given by the outer dimensions of the matrices being multiplied.
• If ( A ) is an ( m \times n ) matrix and ( B ) is an ( n \times p ) matrix, the product ( C = A \times B ) will be an ( m \times p ) matrix.

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Matrix Multiplication Example (1x3 and 3x2)

$$\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\times \left[\begin{array}{cc}4& 5\\ 6& 7\\ 8& 9\end{array}\right]=\left[\begin{array}{cc}1\cdot 4+2\cdot 6+3\cdot 8& 1\cdot 5+2\cdot 7+3\cdot 9\end{array}\right]=\left[\begin{array}{cc}40& 46\end{array}\right]$$ $$\text{Let }A\text{ be a matrix of size }m\times n\text{ and let }B\text{ be a matrix of size }p\times q.$$ $$\text{The product }AB\text{ is defined if and only if }n=p.$$ $$\text{The resulting matrix }AB\text{ will have the dimensions }m\times q.$$ $$\text{Example 1: Possible Multiplication}$$ $$\text{Let }A\text{ be a }2\times 3\text{ matrix and let }B\text{ be a }3\times 2\text{ matrix.}$$ $$A=\left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right),B=\left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)$$ $$\text{Since the number of columns in }A\text{ (which is 3) equals the number of rows in }B\text{ (which is 3), the product }AB\text{ is defined.}$$ $$\text{The resulting matrix }AB\text{ will be a }2\times 2\text{ matrix.}$$ $$\text{Example 2: Impossible Multiplication}$$ $$\text{Let }C\text{ be a }2\times 3\text{ matrix and let }D\text{ be a }2\times 2\text{ matrix.}$$ $$C=\left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right),D=\left(\begin{array}{cc}g& h\\ i& j\end{array}\right)$$ $$\text{Since the number of columns in }C\text{ (which is 3) does not equal the number of rows in }D\text{ (which is 2), the product }CD\text{ is not defined.}$$