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Questions and Answers
What are matrices?
Matrices are rectangular arrays of numbers, symbols, or characters arranged in rows and columns.
What is the order of a matrix defined by?
Matrix algebra is adaptable to a systematic method of mathematical treatment and well suited to ____________.
computers
A square matrix has an equal number of rows and columns.
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Match the following types of matrices with their descriptions:
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What is the condition for two matrices to be equal?
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For matrix addition, matrices of different sizes _____ be added or subtracted.
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Define scalar multiplication of matrices.
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The inverse of a matrix must satisfy the property A * A^(-1) = A^(-1) * A = I.
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Match the matrix operation with its correct description:
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Study Notes
Here are the study notes for the text:
- Matrices and Determinants*
Introduction to Matrices
- A matrix is a rectangular array of numbers, symbols, or characters
- Elements of a matrix are arranged in rows and columns
- Each element is identified by its row and column index
Types of Matrices
- Singleton Matrix: a matrix with only one element (1 × 1)
- Row Matrix: a matrix with only one row (1 × n)
- Column Matrix: a matrix with only one column (m × 1)
- Rectangular Matrix: a matrix with a different number of rows and columns (m × n)
- Square Matrix: a matrix with equal number of rows and columns (n × n)
- Diagonal Matrix: a square matrix with all non-zero elements on the main diagonal
- Unit or Identity Matrix: a diagonal matrix with all main diagonal elements equal to 1
- Null or Zero Matrix: a matrix with all elements equal to 0
- Triangular Matrix: a square matrix with all elements above or below the main diagonal equal to 0
- Upper Triangular Matrix: a triangular matrix with all elements below the main diagonal equal to 0
- Lower Triangular Matrix: a triangular matrix with all elements above the main diagonal equal to 0
- Scalar Matrix: a diagonal matrix with all main diagonal elements equal to the same scalar
Matrix Operations
-
Addition of Matrices: only possible for matrices of the same order
- Two matrices can be added element-wise to form a new matrix
- Commutative and associative laws hold
- Subtraction of Matrices: similar to addition, but with subtraction instead of addition
-
Scalar Multiplication of Matrices: multiplying a matrix by a scalar
- Properties: distributive law, associative law, and scalar multiplication holds
-
Matrix Multiplication: only possible for matrices where the number of columns of the first matrix equals the number of rows of the second matrix
- Properties: associative law, distributive law, and scalar multiplication holds
Equality of Matrices
- Two matrices are equal if all corresponding elements are equal
- Some properties of equality: symmetric, transitive, and reflexive
Transpose of a Matrix
- Interchange rows and columns to form the transpose of a matrix
- Properties of transposed matrices:
- (A+B)T = AT + BT
- (AB)T = BT AT
- (kA)T = kAT
- (AT)T = A### Matrices - Operations
- (A + B)T = AT + BT
- (AB)T = BT AT
Symmetric Matrices
- A square matrix is symmetric if it is equal to its transpose: A = AT
- When the original matrix is square, transposition does not affect the elements of the main diagonal
Identity, Diagonal, and Scalar Matrices
- The identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal to their transpose since the diagonal is unaffected
Inverse of a Matrix
- The inverse of a scalar k is the reciprocal or division of 1 by the scalar: k-1 = 1/k
- The inverse of a square matrix, A, if it exists, is the unique matrix A-1 where: A A-1 = A-1 A = I
- Properties of the inverse:
- (AB) -1 = B-1 A-1
- (A -1) T = A T -1
- (A -1) -1 = A
- (kA) -1 = A -1 / k
Matrix Properties
- A square matrix that has an inverse is called a nonsingular matrix
- A matrix that does not have an inverse is called a singular matrix
- Square matrices have inverses except when the determinant is zero
- When the determinant of a matrix is zero, the matrix is singular
Matrix Operations
- The addition or subtraction of any two matrices is possible if the order of the two matrices is the same
- Multiplication of any two matrices is possible only when the number of columns in the first matrix equals the number of rows in the second matrix
- If the order of a matrix is "m × n", then its transpose matrix will be "n × m", where a transpose matrix is formed by changing the rows of a matrix into columns and its columns into rows
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Description
Solve problems on matrices, determinants, linear dependence, and Gaussian elimination. Learn about rank of a matrix, inverse of matrices, and characteristic polynomials.
