Linear Algebra – Matrix Algebra - Quick Guides

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Study GuideLinear Algebra–Matrix Algebra1.MatricesLinear algebra is built around one very important idea:matrices.A matrix (plural: matrices) is simply arectangular arrangement of numbers. Think of it like numbersneatly organized inrows and columns—similar to a table or spreadsheet.What Is a Matrix?A matrix is written insidelarge parenthesesorsquare brackets.Size (Dimensions) of a MatrixEvery matrix has asize, also called itsdimensions.The size tells us:•How manyrowsit has•How manycolumnsit hasIf a matrix has:•m rows•n columnsWe call it anm × n matrix(read as “m by n”).For example, the matrix shown above has:•2 rows•3 columnsSo, it is a2 × 3 matrix.

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Study GuideImportant:•Rows are countedfrom top to bottom.•Columns are countedfrom left to right.What Are Entries?The individual numbers inside a matrix are calledentries.Each entry has a specific position. We describe its position using:•Therow number first•Thecolumn numbersecondThis is called the(i, j) entry.For example:•The number–2is in row 2, column 1.So it is the(2, 1) entry.•The number0is the(1, 2) entry.•The number1is the(2, 3) entry.So whenever we see something written like:It means:•i =row number•j = column numberMatrix NotationWe usually name matrices using capital letters like:

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Study GuideIf we write:This means:•A is a matrix•The entry in row i and column j is written as (aij)So if A is an m × n matrix, then:•It has m rows•It has n columns•Each entry is written as (aij)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 entriesQuestion:IfHow many entries does Acontain?Solution:Since A is a2 × 3 matrix, it has:

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Study Guide2 rows × 3 columns =6 entriesSo A contains6 numbersin total.Example 2:Finding a Matrix from a FormulaSuppose we are given a 2 × 2 matrix(B)where each entry is defined by:Our goal is tofind the actual matrixby calculating each entry one by one.Step 1:Find Each EntryRemember:•i = row number•j = column numberNow compute each entry carefully.(1,1) entry:(1,2) entry:(2,1) entry:

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Study Guide(2,2) entry:Final MatrixSo the matrix B is:Example 3:The Kronecker Delta MatrixNow we are asked to build a 3 × 3 matrix using this rule:This symbolδijis called theKronecker delta.What Does This Rule Mean?•If the row number equals the column number → write1•If the row number is different from the column number → write0So:•(1,1) = 1•(2,2) = 1

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Study Guide•(3,3) = 1All other entries = 0The Matrix Looks Like ThisThis matrix is called theidentity matrix of order 3.Entries Along the DiagonalAn entry is called adiagonal entryif:Row number = Column numberExamples:•(1,1)•(2,2)•(3,3)All other entries are calledoff-diagonal entries.Important Observations•In matrix A, diagonal entries are (a11) and (a22).•In matrix B, diagonal entries are (b11) and (b22).•In the Kronecker delta matrix, diagonal entries are all 1.Diagonal MatrixIfevery off-diagonal entry is zero, the matrix is called adiagonal 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 MatricesA matrix is called asquare matrixif:Number of rows = Number of columnsSo its size is:We say the matrix hasorder n.Examples:•2 × 2 matrix → order 2•3 × 3 matrix → order 3Triangular MatricesNow let’s look at special types of square matrices.UpperTriangular MatrixIfall entries below the main diagonal are zero, the matrix is calledupper triangular.That means:•Everything below the diagonal = 0•Numbers can appear on or above the diagonalLower Triangular MatrixIfall entries above the main diagonal are zero, the matrix is calledlower triangular.That means:

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Study Guide•Everything above the diagonal = 0•Numbers can appear on or below the diagonalImportant NoteA matrix is calledtriangularif it is:•Upper triangular, or•Lower triangularA diagonal matrix is special because it is:•Both upper triangular•And lower triangularTranspose of a MatrixOne very important operation in linear algebra is thetranspose.If A is a matrix, the transpose of A is written:What Does Transpose Mean?Tofind the transpose:Turn rows into columns.More precisely:•Row 1 becomes Column 1•Row 2 becomes Column 2•And so onIf A is m × n, then:

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Study Guideis n × m.So the dimensions swap.Key PropertyIf you transpose twice, you get back the original matrix:Example 4:Transpose of a MatrixWe are given the 2 × 3 matrix:Step 1:What Does Transpose Mean?To find thetransposeof a matrix:Change rows into columns.That means:•Row 1 becomes Column 1•Row 2 becomes Column 2If 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 TransposeNotice how:

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Study Guide•The first row of A became the first column of (AT)•The second row of A became the second column of (AT)Example 5:Matrices Equal to Their Own TransposeIn earlier examples (Examples 2 and 3), something interesting happened.Those matrices were equal to their transpose.That means:Symmetric MatricesAny matrix that equals its own transpose is called a:Symmetric MatrixSo if:then A is symmetric.Symmetric matrices must always besquare matrices, because only square matrices can be equal totheir transpose.Row and Column MatricesNow let’s look at special types of matrices based on shape.Row MatrixA matrix with exactlyone rowis called arow matrix.Example:

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