Podcast
Questions and Answers
What is true about the inverse of a matrix?
What is true about the inverse of a matrix?
- Inverse of a matrix always exists.
- Inverse of a matrix is not unique.
- Inverse of a matrix, if it exists, is unique. (correct)
- Inverse of a matrix is always a square matrix.
What is a necessary condition for the product BA to be equal to the product AB?
What is a necessary condition for the product BA to be equal to the product AB?
- Matrices A and B should be of the same order.
- Matrices A and B should be rectangular matrices of the same order.
- Matrices A and B should be square matrices of the same order. (correct)
- Matrices A and B should be diagonal matrices of the same order.
What is the inverse of a rectangular matrix?
What is the inverse of a rectangular matrix?
- It always exists.
- It never exists. (correct)
- It exists only for some rectangular matrices.
- It exists for all rectangular matrices except one.
If B is the inverse of A, what is true about A?
If B is the inverse of A, what is true about A?
What is the formula for the inverse of the product of two invertible matrices?
What is the formula for the inverse of the product of two invertible matrices?
What can be said about the matrices X, A and B if X = AB?
What can be said about the matrices X, A and B if X = AB?
Why is the inverse of a matrix, if it exists, unique?
Why is the inverse of a matrix, if it exists, unique?
What is a prerequisite for the inverse of a matrix to exist?
What is a prerequisite for the inverse of a matrix to exist?
What is the correct statement about the inverse of a matrix?
What is the correct statement about the inverse of a matrix?
What can be said about the matrices A and B if AB = I?
What can be said about the matrices A and B if AB = I?
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Study Notes
• The concept of matrices is essential in various branches of mathematics, and it simplifies work to a great extent compared to other straightforward methods.
• Matrices are not only used to represent coefficients in systems of linear equations but also have far-reaching utility in electronic spreadsheet programs, business, science, genetics, economics, sociology, modern psychology, and industrial management.
• A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
• A scalar matrix is a diagonal matrix where all diagonal elements are equal, and it can be denoted as B = [bij] n × n, where bij = 0 when i ≠j, and bij = k when i = j, for some constant k.
• An identity matrix is a square matrix where all elements in the diagonal are 1, and the rest are 0, and it can be denoted as A = [aij] n × n, where aij = 1 if i = j, and aij = 0 if i ≠j.
• A zero matrix is a matrix where all elements are 0, and it can be denoted as O.
• The negative of a matrix A is denoted by –A, and it is defined as –A = (–1) A.
• The difference of two matrices A and B of the same order is defined as A – B = A + (–1) B.
• The addition of matrices satisfies the following properties: commutative law (A + B = B + A), associative law (A + (B + C) = (A + B) + C), and additive identity (A + O = A).
• The properties of matrix addition include the commutative law, associative law, and additive identity.
• A matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.
• Elementary operations or transformations on a matrix include three operations due to rows and three due to columns, which are interchange of any two rows or columns, multiplication of a row or column by a non-zero number, and addition of a row or column to another row or column multiplied by a number.
• The inverse of a matrix A is denoted by A^–1, and it satisfies the property AA^–1 = A^–1A = I.
• If B is the inverse of A, then A is also the inverse of B, and A^–1 = B.
• The inverse of a square matrix, if it exists, is unique.
• If A and B are invertible matrices of the same order, then (AB)^–1 = B^–1A^–1.
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