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Lecture 8 System of Linear Differential Equations fall semester
Instructor: A. S. Brwa / MSc. In Structural Engineering College of Engineering / Ishik University
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Ishik University Review of Matrix Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. Matrices with more than one row and one column denoted by bold, upper case letters (e.g., X). π π π βπ π π Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Look at this Matrix:
Ishik University Review of Matrix Look at this Matrix: These are called elements or entries. Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Ishik University
Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix What's the size of this matrix?
Ishik University Review of Matrix What's the size of this matrix? 3Β rows andΒ 2Β columns. This is aΒ 3 by 2Β matrix (3x2). Note: AΒ 2 x 3Β matrix and aΒ 3 x 2Β matrix are definitely different sizes! Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Ishik University Review of Matrix Sometimes, we'll need to refer to a specific entry, so we have a special "tagging" system.Β It's based on rows and columns: This element is located in the row and in the column Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Ishik University
Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Ishik University Review of Matrix A vector: is a special type of matrix that has only one row (called a row vector) or one column (called a column vector). Below, a is a column vector while b is a row vector. Vectors are denoted by bold, lower case letters (e.g., x). Row vector Column vector A scalar: is a matrix with only one row and one column. It is customary to denote scalars by italicized, lower case letters (e.g., x) Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Ishik University Review of Matrix Square Matrices: A square matrix A has the same number of rows and columns. That is, A is n x n. In this case, A is said to have order n. Faculty of Engineering β Differential Equations β Lecture 8β System of Linear Differential Equations
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Ishik University Review of Matrix The Zero Matrix: The zero matrix is defined to be 0 = (0), whose dimensions depend on the context. For example, Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Ishik University Review of Matrix Identity matrix: is a diagonal matrix with 1βs and only 1βs on the diagonal and zeros everywhere else. The identity matrix is almost always denoted as I. Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Adding and Subtracting Matrices\
Ishik University Review of Matrix Adding and Subtracting Matrices\ Example: Subtract Matrix A and B Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Adding and Subtracting Matrices\
Ishik University Review of Matrix Adding and Subtracting Matrices\ Adding and subtracting matrix is really easy, but THE SIZES MUST BE THE SAME! Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Adding and Subtracting Matrices\
Ishik University Review of Matrix Adding and Subtracting Matrices\ Adding and subtracting matrix is really easy, but THE SIZES MUST BE THE SAME! Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Adding and Subtracting Matrices\
Ishik University Review of Matrix Adding and Subtracting Matrices\ Example: Add Matrix A and B Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication by scalar, A scalar is just a number likeΒ 3Β orΒ -5Β orΒ 2 7 Β Β orΒ .4. We can multiply a matrix by some value: πΊπππππ(π)= π ππ§π Matrix π= π π π βπ π π πβπ¨=π π π π βπ π π Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication by scalar. Example: If Matrix FindΒ 2A... (2Β timesΒ A) Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication by scalar. Β Example: πΌπ πππ‘πππ₯ π΄= π π π π π π Find 5A. Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication by scalar. Β Example: Now, letβs try more difficult oneΒ π°π π΄πππππ π¨= π π π π π π πππ
π©= π π π π π ππ , ππ’π§π ππβπ. Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication by scalar. Β Example: Now, letβs try more difficult oneΒ π°π π΄πππππ π¨= π π π π π π πππ
π©= π π π π π ππ , ππ’π§π ππβπ. Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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And the answer will be a 3 x 1 matrix
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication of two matrices. Β ToΒ multiply two matrices togetherΒ is a bit more difficult. First of all, the size of the two matrices you are multiplying is super important! If the size of matrixΒ AΒ isΒ 3 x 4 and the size of matrixΒ BΒ isΒ 4 x 1 And the answer will be a 3 x 1 matrix Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication of two matrices. Β Example: If Matrix π¨= βπ π π βπ πππ
π© π π π π ππππ
π¨βπ© Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Kurdistan Technical Institute (KTI)
Ishik University Review of Matrix π¨= βπ π π βπ πππ
π© π π π π ππππ
π¨βπ© A is 2 by 2 and B is 2 by 2. So, the result is The result is 2 by 2 23/27 Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations Department of Petroleum Refining
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Kurdistan Technical Institute (KTI)
Ishik University Review of Matrix π¨= βπ π π βπ πππ
π© π π π π ππππ
π¨βπ© A is 2 by 2 and B is 2 by 2. So, the result is The result is 2 by 2 2 rows and 2 columns 24/27 Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations Department of Petroleum Refining
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Review of Matrix Multiplication of Matrices\
Ishik University Review of Matrix Multiplication of Matrices\ Multiplication of two matrices. Β Example: If Matrixπ¨= π π π βπ π π πππ
π© βπ π π βπ ππππ
π¨βπ© Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Kurdistan Technical Institute (KTI)
Ishik University Review of Matrix π¨= π π π βπ π π πππ
π© βπ π π βπ ππππ
π¨βπ© 26/27 Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations Department of Petroleum Refining
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Kurdistan Technical Institute (KTI)
Ishik University Review of Matrix π¨= π π π βπ π π πππ
π© βπ π π βπ ππππ
π¨βπ© 27/27 Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations Department of Petroleum Refining
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Ishik University Review of Matrix Determinant Matrix The determinant of a matrix is aΒ special numberΒ that can be calculated from a square matrix. Its very helpful in solving the system of linear equations and inverse matrix Example: If matrix Then, its determinate |π¨| = ππ βππ Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Determinant Matrix Example: If matrix A =
Ishik University Review of Matrix Determinant Matrix Example: If matrix A = Then, its determinate |A| = 3Γ6 β 8Γ4 = 18 β 32 =Β β14 Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Determinant Matrix Example: If matrix
Ishik University Review of Matrix Determinant Matrix Example: If matrix Then, its determinate π© = πΓπ β πΓπ = ππβππ = ππ Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Determinant Matrix Example: If matrix
Ishik University Review of Matrix Determinant Matrix Example: If matrix The determinant is; |π¨| = π(ππ β ππ) β π(π
π β ππ) + π(π
π β ππ) Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Determinant Matrix Example: If matrix
Ishik University Review of Matrix Determinant Matrix Example: If matrix Then, its determinate πͺ = 6Γ β2Γ7 β 5Γ8 β 1Γ 4Γ7 β 5Γ2 +1Γ 4Γ8 β β2Γ2 = 6Γ β54 β 1Γ Γ 36 = βπππ Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Inverse Matrix Example: If matrix Find Inverse of A
Ishik University Review of Matrix Inverse Matrix Example: If matrix Find Inverse of A Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Inverse Matrix Example: If matrix
Ishik University Review of Matrix Inverse Matrix Example: If matrix First need to find determinants of A Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Inverse Matrix First need to find determinants of A
Ishik University Review of Matrix Inverse Matrix First need to find determinants of A If determinant of A is NOT zero, then matrix A is invertible (non-singular). If determinant of A is zero, then matrix A is NOT invertible (singular). Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Second need to find matrix of cofactors
Ishik University Review of Matrix Second need to find matrix of cofactors Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Third Step: Adjugate (also called Adjoint)
Ishik University Review of Matrix Third Step: Adjugate (also called Adjoint) Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same): Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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Review of Matrix Forth Step: Multiply by 1/Determinant
Ishik University Review of Matrix Forth Step: Multiply by 1/Determinant Faculty of Engineering β Differential Equations β Lecture 8 β System of Linear Differential Equations
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