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What is the result of the matrix addition A + B for A = [[3, 5], [1, 4]] and B = [[2, -1], [0, 3]]?
What is the result of the matrix addition A + B for A = [[3, 5], [1, 4]] and B = [[2, -1], [0, 3]]?
[[5, 4], [1, 7]]
Compute the result of scalar multiplication for 3A, where A = [[2, 1], [-1, 4]].
Compute the result of scalar multiplication for 3A, where A = [[2, 1], [-1, 4]].
[[6, 3], [-3, 12]]
Calculate the product AB for A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]].
Calculate the product AB for A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]].
[[4, 6], [10, 12]]
Find the determinant of matrix A = [[5, 3], [-2, 4]] using the formula det(A) = ad - bc.
Find the determinant of matrix A = [[5, 3], [-2, 4]] using the formula det(A) = ad - bc.
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What is the inverse of the matrix A = [[1, 2], [3, 4]] if it exists?
What is the inverse of the matrix A = [[1, 2], [3, 4]] if it exists?
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Use the matrix method to find the values of x and y for the equations 2x + 3y = 13 and x - 2y = -3.
Use the matrix method to find the values of x and y for the equations 2x + 3y = 13 and x - 2y = -3.
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Determine if the matrix A = [[2, 4], [1, 2]] is singular.
Determine if the matrix A = [[2, 4], [1, 2]] is singular.
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What is the adjoint of the matrix A = [[1, 3], [2, 4]]?
What is the adjoint of the matrix A = [[1, 3], [2, 4]]?
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Verify the property det(AB) = det(A) * det(B) for A = [[1, 0], [0, 1]] and B = [[2, 3], [4, 5]].
Verify the property det(AB) = det(A) * det(B) for A = [[1, 0], [0, 1]] and B = [[2, 3], [4, 5]].
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Solve for x and y in the cost problem represented by the equations 2x + 3y = 50 and x + 2y = 30 using matrices.
Solve for x and y in the cost problem represented by the equations 2x + 3y = 50 and x + 2y = 30 using matrices.
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Study Notes
Matrix Operations
- Matrix Addition/Subtraction: Add/subtract corresponding elements of matrices with same dimensions.
- Scalar Multiplication: Multiply every element of a matrix by a scalar (number).
- Matrix Multiplication: Multiply matrices element-wise, following the rule: row times column.
- Determinant: A value calculated from a square matrix, used in finding inverses and solving other problems. Determined using the formula det(A) = ad - bc for a 2x2 matrix.
- Inverse of a Matrix: A matrix multiplied by its inverse yields the identity matrix. The inverse of a 2x2 matrix is calculated using the formula A-1 = (1/det(A)) * [[d, -b], [-c, a]].
Systems of Equations
- Representing with Matrices: A system of equations can be written as a matrix equation: AX = B.
- Solving using Matrix Inverses: The solution for X in AX = B is given by X = A-1B.
Singular Matrices
- Singular Matrix: A matrix that does not have an inverse, characterized by a determinant of 0.
Adjoint of a Matrix
- Adjoint: A specific mathematical operation on a matrix. Adj(A) = [[d, -b], [-c, a]] for a 2x2 matrix.
Verification of Matrix Properties
- Property to Verify: The determinant of a product of matrices equals the product of the determinants of each of the individual matrices det(AB) = det(A) * det(B).
Application Problem (Cost of Pens and Notebooks)
- Representing a Problem with Matrices: Problem involving costs of pens and notebooks represented through a system of equations can be represented as a matrix equation: AX = B.
- Solving for Unknown Variables: Involve solving for unknowns using matrix inverses.
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Description
Explore fundamental concepts of Matrix Operations including addition, subtraction, scalar multiplication, multiplication, determinants, and inverses. Additionally, learn how to represent and solve systems of equations using matrix methods. Test your understanding of these key topics in linear algebra.
