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Linear Algebra – Matrix Algebra

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    Study Guide Linear Algebra – Matrix Algebra 1. Matrices Linear algebra is built around one very important idea: matrices . A matrix (plural: matrices) is simply a rectangular arrangement of numbers . Think of it like numbers neatly organized in rows and columns — similar to a table or spreadsheet. What Is a Matrix? A matrix is written inside large parentheses or square brackets . Size (Dimensions) of a Matrix Every matrix has a size , also called its dimensions . The size tells us: • How many rows it has • How many columns it has If a matrix has: • m rows • n columns We call it an m × n matrix (read as “m by n”). For example, the matrix shown above has: • 2 rows • 3 columns So, it is a 2 × 3 matrix .

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    Study Guide Important: • Rows are counted from top to bottom . • Columns are counted from left to right . What Are Entries? The individual numbers inside a matrix are called entries . Each entry has a specific position. We describe its position using: • The row number first • The column number second This is called the (i, j) entry . For example: • The number – 2 is in row 2, column 1. So it is the (2, 1) entry . • The number 0 is the (1, 2) entry . • The number 1 is the (2, 3) entry . So whenever we see something written like: It means: • i = row number • j = column number Matrix Notation We usually name matrices using capital letters like:

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    Study Guide If we write: This means: • A is a matrix • The entry in row i and column j is written as (a ij ) So if A is an m × n matrix, then: • It has m rows • It has n columns • Each entry is written as (a ij ) Example 1 : The set of all m × n matrices with real number entries is written as: This means: • All matrices with m rows • n columns • Real numbers as entries Question: If How many entries does A contain? Solution: Since A is a 2 × 3 matrix , it has:

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    Study Guide 2 rows × 3 columns = 6 entries So A contains 6 numbers in total. Example 2: Finding a Matrix from a Formula Suppose we are given a 2 × 2 matrix ( B ) where each entry is defined by: Our goal is to find the actual matrix by calculating each entry one by one. Step 1: Find Each Entry Remember: • i = row number • j = column number Now compute each entry carefully. (1,1) entry: (1,2) entry: (2,1) entry:

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    Study Guide (2,2) entry: Final Matrix So the matrix B is: Example 3: The Kronecker Delta Matrix Now we are asked to build a 3 × 3 matrix using this rule: This symbol δ ij is called the Kronecker delta . What Does This Rule Mean? • If the row number equals the column number → write 1 • If the row number is different from the column number → write 0 So: • (1,1) = 1 • (2,2) = 1

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    Study Guide • (3,3) = 1 All other entries = 0 The Matrix Looks Like This This matrix is called the identity matrix of order 3 . Entries Along the Diagonal An entry is called a diagonal entry if: Row number = Column number Examples: • (1,1) • (2,2) • (3,3) All other entries are called off - diagonal entries . Important Observations • In matrix A, diagonal entries are (a 11 ) and (a 22 ). • In matrix B, diagonal entries are (b 11 ) and (b 22 ). • In the Kronecker delta matrix, diagonal entries are all 1. Diagonal Matrix If every off - diagonal entry is zero , the matrix is called a diagonal matrix . That means:

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    Study Guide • Only the diagonal can contain nonzero numbers. • Everything else must be 0. Sometimes blank spaces are left instead of writing zero — especially in large matrices. Square Matrices A matrix is called a square matrix if: Number of rows = Number of columns So its size is: We say the matrix has order n . Examples: • 2 × 2 matrix → order 2 • 3 × 3 matrix → order 3 Triangular Matrices Now let’s look at special types of square matrices. Upper Triangular Matrix If all entries below the main diagonal are zero , the matrix is called upper triangular . That means: • Everything below the diagonal = 0 • Numbers can appear on or above the diagonal Lower Triangular Matrix If all entries above the main diagonal are zero , the matrix is called lower triangular . That means:

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    Study Guide • Everything above the diagonal = 0 • Numbers can appear on or below the diagonal Important Note A matrix is called triangular if it is: • Upper triangular, or • Lower triangular A diagonal matrix is special because it is: • Both upper triangular • And lower triangular Transpose of a Matrix One very important operation in linear algebra is the transpose . If A is a matrix, the transpose of A is written: What Does Transpose Mean? To find the transpose: Turn rows into columns. More precisely: • Row 1 becomes Column 1 • Row 2 becomes Column 2 • And so on If A is m × n, then:

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    Study Guide is n × m. So the dimensions swap. Key Property If you transpose twice, you get back the original matrix: Example 4: Transpose of a Matrix We are given the 2 × 3 matrix: Step 1: What Does Transpose Mean? To find the transpose of a matrix: Change rows into columns. That means: • Row 1 becomes Column 1 • Row 2 becomes Column 2 If a matrix is m × n, its transpose will be n × m. So since A is 2 × 3, its transpose will be 3 × 2. Step 2: Write the Transpose Notice how:

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    Study Guide • The first row of A became the first column of (A T ) • The second row of A became the second column of (A T ) Example 5: Matrices Equal to Their Own Transpose In earlier examples (Examples 2 and 3), something interesting happened. Those matrices were equal to their transpose. That means: Symmetric Matrices Any matrix that equals its own transpose is called a: Symmetric Matrix So if: then A is symmetric. Symmetric matrices must always be square matrices , because only square matrices can be equal to their transpose. Row and Column Matrices Now let’s look at special types of matrices based on shape. Row Matrix A matrix with exactly one row is called a row matrix . Example:

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