2.1 Matrix Algebra

• What is a matrix defined as?

  Ordered array of numbers
• The dimensions of a matrix are described as rows × columns.
• A matrix is a three-dimensional array of numbers.

  False
• What is matrix notation used for?

  Identifying matrix elements
• In the general format A = [a_{ij}]</latex>, the index i represents the row number.
• $a_{ij}$ denotes the element at the i-th column and j-th row.


  False
• What condition must two matrices satisfy to be added?

  Same dimensions
• Matrix addition is commutative, meaning A + B = B + A.
• How is matrix subtraction performed?

  Subtract corresponding elements
• Matrix subtraction requires both matrices to have the same dimensions.
• Steps of the scientific method

  1️⃣ Make an observation

  2️⃣ Form a hypothesis

  3️⃣ Test the hypothesis with an experiment

  4️⃣ Analyze the data

  5️⃣ Draw a conclusion
• What is the formula for matrix subtraction?

  $C =$$A - B$
• A matrix is an ordered rectangular array of numbers arranged in rows and columns
• The dimensions of a matrix are described as rows × columns.
• What does $a_{ij}$ represent in matrix notation?


  Element at row i, column j
• Two matrices must have the same dimensions to be added.
• The element $c_{ij}$ in matrix addition is calculated as c_{ij} = a_{ij} + b_{ij}</latex> for all i and j
• What is the formula for matrix subtraction?

  $C =$$A - B$
• Scalar multiplication involves multiplying each element in the matrix by the scalar.
• If $A =$$[a_{ij}]$ and $k$ is a scalar, then $kA =$$[ka_{ij}]$ for all i and j
• What condition must the dimensions of two matrices satisfy for multiplication?

  Columns of A = rows of B
• The element $c_{ij}$ in matrix multiplication is calculated as $c_{ij} =$$\sum_{k = 1}^{n} a_{ik} \times b_{kj}$, where $n$ is the number of columns in A and rows in B
• What is the non-commutativity property of matrix multiplication?

  $A \times B \neq B \times A$
• Matrix multiplication is non-commutative, meaning $A \times B \neq B \times A$ in general
• Matrix multiplication is associative, meaning (A \times B) \times C = A \times (B \times C)</latex>
• Matrix multiplication is distributive over addition
• The identity matrix $I$ satisfies $A \times I =$$A =$$I \times A$.
• What is the term for an ordered rectangular array of numbers arranged in rows and columns?

  Matrix
• The dimensions of a matrix are described as rows × columns
• In matrix notation $A =$$[a_{ij}]$, the term $a_{ij}$ represents the element at the i-th row and j-th column.
• To add two matrices, they must have the same dimensions
• When adding two matrices $A$ and $B$, the resulting element $c_{ij}$ is calculated as $c_{ij} =$$a_{ij} +$$b_{ij}$.
• What is the sum of the matrices A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}</latex> and $B =$$\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$?


  $\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$
• To add two matrices, they must have the same dimensions
• What is the sum of two matrices A and B with dimensions $m \times n$?


  $C =$$A +$$B$
• What is the result of subtracting two $m \times n$ matrices A and B?


  $C =$$A - B$
• Scalar multiplication involves multiplying a matrix by a single number
• If $A =$$[a_{ij}]$ and $k$ is a scalar, what is $kA$?


  $kA =$$[ka_{ij}]$
• The number of columns in the first matrix must equal the number of rows in the second matrix for matrix multiplication.
• In matrix multiplication, the element $c_{ij}$ is calculated as c_{ij} = \sum_{k = 1}^{n} a_{ik} \times b_{kj}</latex>, where $n$ is the number of columns