MATH1131 Chap.6 Matrix Algebra, Inverses and Determinants
Matrix Algebra, Inverses and Determinants
Matrices package linear systems into single objects. The product AB is defined when the inner sizes agree, its entries are dot products of rows of A with columns of B, and it is generally not equal to BA, so the order of factors matters and cancellation is not automatic. The transpose swaps rows and columns and reverses the order of products.
A square matrix A is invertible when some matrix B satisfies AB = BA = I; the inverse is found by row-reducing the augmented matrix of A and the identity, and a 2 by 2 inverse has a direct formula involving ad - bc. The determinant decides invertibility, since A is invertible exactly when det A is non-zero, and it also measures how A scales area or volume, with sign recording orientation.
Determinants are computed by cofactor expansion along a convenient row or column or, more efficiently, by row reduction to triangular form while tracking how each operation changes the value. Key properties include that a matrix and its transpose share a determinant and that det(AB) equals det A multiplied by det B, and they connect directly back to whether Ax = b has a unique solution.
What this chapter covers
- 01
Matrix addition, scalar multiples and products
- 02
Non-commutativity and the zero-product warning
- 03
Transposes and their properties
- 04
Inverses by row reduction and the 2 by 2 formula
- 05
Determinants by cofactor expansion
- 06
Row operations and determinants
- 07
Products, transposes and inverses of determinants
- 08
Determinants and solutions of Ax = b
Worked example · free
A determinant through row operations
- 1R2 - 2R1 gives (0, 1, 1) and R3 + R1 gives (0, 5, 6); these operations leave the determinant unchanged.
- 1Expand down the first column: det M = 1 times det [[1, 1], [5, 6]].
- 1Compute the 2 by 2 determinant: 1(6) - 1(5) = 1, so det M = 1.
- 1Since det M is non-zero, M is invertible and Mx = b has exactly one solution for every b.
Key terms
- Identity matrix
- The square matrix with ones on the main diagonal and zeros elsewhere, which leaves any compatible matrix unchanged under multiplication.
- Inverse matrix
- A matrix whose product with the given square matrix in either order is the identity.
- Transpose
- The matrix obtained by turning rows into columns, so the (i, j) entry moves to position (j, i).
- Determinant
- A number assigned to a square matrix that is non-zero exactly when the matrix is invertible.
- Cofactor expansion
- A method that computes a determinant as a signed sum of entries times smaller determinants along one row or column.
Matrix Algebra, Inverses and Determinants FAQ
Why is AB usually different from BA?
Each entry of AB combines rows of A with columns of B, and swapping the factors pairs different rows and columns. Even when both products exist and have the same size, the entries generally differ, as a 2 by 2 example quickly shows.
What is the fastest way to find a 3 by 3 inverse?
Row-reduce the augmented matrix of A and the identity until the left block becomes the identity; the right block is then the inverse. Check the answer by multiplying it with A.
How does scaling a matrix affect its determinant?
Multiplying an n by n matrix by c multiplies every row by c, so the determinant is multiplied by c to the power n. For a 3 by 3 matrix, doubling it multiplies the determinant by eight.
What does a zero determinant mean for a linear system?
The coefficient matrix is not invertible, so Ax = b never has a unique solution: depending on b it has no solution or infinitely many. Row reduction decides which.
How do I check a matrix inverse quickly?
Multiply your answer by the original matrix and confirm that the product is the identity. Checking one row and one column first catches most arithmetic slips before you complete the full product.
Exam move
Learn the properties of transposes, inverses and determinants as a compact list, and for each property test a small example yourself to fix it in memory. Practise both routes to a determinant, cofactor expansion and row reduction, and use one to check the other. When finding inverses, always multiply back.
Connect this chapter to the previous one by solving a system both by elimination and by an inverse, and by predicting from the determinant whether a unique solution exists before doing any reduction. Before computing a large determinant, scan for a row or column with several zeros, or for two proportional rows, since either can shorten the work to a single line.
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