Aussie Maths & Science Teachers: Save your time with SmarterEd
How many of the following statements are true?
\(D\)
\(\text{All square matrices have an inverse → not true}\)
\(\text{The inverse of a matrix could be the same as the transpose of that matrix → true}\)
\(\text{If the determinant of a matrix is equal to zero, then the inverse does not exist → true}\)
\(\text{It is possible to take the inverse of an identity matrix → true}\)
\(\Rightarrow D\)
If matrix `M = [(3,2), (8,9), (13,7)]` , then its transpose, `M^T` , is
| A. `[(2,3), (9,8), (7,13)]` | B. `[(2,9,7), (3,8,13)]` | |
| C. `[(7,9,2), (13,8,3)]` | D. `[(3,8,13), (2,9,7)]` | |
| E. `[(13,8,3), (7,9,2)]` |
`D`
`text{Transpose} \ x_{ij} -> x_{ji}`
`[(3,2), (8,9), (13,7)]^T = [(3,8,13), (2,9,7)]`
`=>D`
The table below shows information about three matrices: `A, B` and `C`.
| `qquad qquad text(Matrix) qquad qquad` | `quad qquad qquad text(Order) quad qquad qquad` | |
| `A` | `2 xx 4` | |
| `B` | `2 xx 3` | |
| `C` | `3 xx 4` |
The transpose of matrix `A`, for example, is written as `A^T`.
What is the order of the product `C^T xx (A^T xx B)^T`?
`E`
`C^T xx (A^T xx B)^T`
`= (3 xx 4)^T xx ((2 xx 4)^T xx (2 xx 3))^T`
`= (4 xx 3) xx ((4 xx 2) xx (2 xx 3))^T`
`= (4 xx 3) xx (4 xx 3)^T`
`= (4 xx 3) xx (3 xx 4)`
`= 4 xx 4`
`=> E`
In matrix `A`, the element `a_21 = 7`.
The order of matrix `A` is `3 xx 2`.
Matrix `B` is the transpose of matrix `A`.
Matrix `B` could be
| A. | `\ [(3,9,4),(2,7,1)]` | B. | `\ [(4,7,2),(1,0,5)]` | C. | `\ [(4,1,3),(7,8,0)]` |
| D. | `\ [(5,6),(7,1),(4,2)]` | E. | `\ [(9,7),(3,4),(0,2)]` |
`B`
`A = (3 xx 2) \ => \ a_21 = 7`
`text(S)text(ince)\ \ B = A^T:`
`B = (2 xx 3) \ => \ b_12=7`
`=>\ B`
The transpose of `[(2,7,10),(13,19,8)]` is
| A. | `[(13,19,8),(2,7,10)]` |
| B. | `[(10,7,2),(8,19,13)]` |
| C. | `[(2,13),(7,19),(10,8)]` |
| D. | `[(13,2),(19,7),(8,10)]` |
| E. | `[(8,10),(19,7),(13,2)]` |
`C`
`text(Transpose is a new matrix where)`
`text(the rows of 1 matrix become the)`
`text(columns.)`
`text(i.e.)\ \ x_(ij) => x_(ji)`
| `[(2,7,10),(13,19,8)]^T` | `= [(2,13),(7,19),(10,8)]` |
`=> C`